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Caraway

Caraway, also known as meridian fennel and Persian cumin (Carum carvi), is a biennial plant in the family Apiaceae, native to western Asia, Europe, and North Africa.

Etymology
The etymology of "caraway" is unclear. Caraway has been called by many names in different regions, with names deriving from the Latin cuminum (cumin), the Greek karon (again, cumin), which was adapted into Latin as carum (now meaning caraway), and the Sanskrit karavi, sometimes translated as "caraway", but other times understood to mean "fennel". English use of the term caraway dates to at least 1440, possibly having Arabic origin.

Description
The plant is similar in appearance to other members of the carrot family, with finely divided, feathery leaves with thread-like divisions, growing on 20–30 cm stems. The main flower stem is 30–60 cm tall, with small white or pink flowers in compound umbels composed of 5–16 unequal rays 1-6 cm long. Caraway fruits, informally called seeds, are smooth, crescent-shaped, laterally compressed achenes, around 3 mm long, with five pale ridges and a distinctive pleasant smell when crushed. It flowers in June and July.

History
Caraway was mentioned by the early Greek botanist Pedanius Dioscorides as a herb and tonic. It was later mentioned in the Roman Apicius as an ingredient in recipes. Caraway was known in the Arab world as karawiya, and cultivated in Morocco.

Cultivation
The only species that is cultivated is Carum carvi, its fruits being used in many ways in cooking and in the preparation of medicinal products and liqueurs.

The plant prefers warm, sunny locations and well-drained soil rich in organic matter. In warmer regions, it is planted in the winter as an annual. In temperate climates, it is planted as a summer annual or biennial.

It is widely established as a cultivated plant. The Netherlands, Poland and Germany are the top caraway producers. Finland supplies about 28% (2011) of the world's caraway production from some 1500 farms, the high output occurring possibly from its favorable climate and latitudes, which ensure long summer hours of sunlight.

Nutrition
Caraway seeds are 10% water, 50% carbohydrates, 20% protein, and 15% fat (table). In a 100 g reference amount, caraway seeds are a rich source (20% or more of the Daily Value, DV) of protein, B vitamins (24–33% DV), vitamin C (25% DV), and several dietary minerals, especially iron (125% DV), phosphorus (81% DV), and zinc (58% DV) (table).

Phytochemicals
When ground, caraway seeds yield up to 7.5% of volatile oil, mostly D-carvone, and 15% fixed oil of which the major fatty acids are oleic, linoleic, petroselinic, and palmitic acids.

Phytochemicals identified in caraway seed oil include thymol, o-cymene, γ‑terpinene, trimethylene dichloride, β-pinene, 2-(1-cyclohexenyl), cyclohexanone, β-phellandrene, 3-carene, α-thujene, and linalool.

Uses
The fruits, usually used whole, have a pungent, anise-like flavor and aroma that comes from essential oils, mostly carvone, limonene, and anethole. Caraway is used as a spice in breads, especially rye bread. A common use of caraway is whole as an addition to rye bread – often called seeded rye or Jewish rye bread (see Borodinsky bread). Caraway seeds are often used in Irish soda bread and other baked goods.

Caraway may be used in desserts, liquors, casseroles, and other foods. Its leaves can be added to salads, stews, and soups, and are sometimes consumed as herbs, either raw, dried, or cooked, similar to parsley. The root is consumed as a winter root vegetable in some places, similar to parsnips.

Caraway fruits are found in diverse European cuisines and dishes, for example sauerkraut, and the United Kingdom's caraway seed cake. In Austrian cuisine, it is used to season beef and, in German cuisine, pork. In Hungarian cuisine, it is added to goulash, and in Norwegian cuisine and Swedish cuisine, it is used for making caraway black bread.

Caraway oil is used to for the production of Kümmel liquor in Germany and Russia, Scandinavian akvavit, and Icelandic brennivín. Caraway can be infused in a variety of cheeses, such as havarti and bondost to add flavor. In Latvian cuisine, whole caraway seeds are added to the Jāņi sour milk cheese. In Oxford, where the plant appeared to have become naturalised in a meadow, the seeds were formerly offered on a tray by publicans to people who wished to disguise the odour of their drinker's breath.

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title: Caraway


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Cayley–Hamilton theorem

In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies its own characteristic equation.

The characteristic polynomial of an $n × n$ matrix $A$ is defined as $$p_A(\lambda)=\det(\lambda I_n-A)$$, where $det$ is the determinant operation, $λ$ is a variable scalar element of the base ring, and $I_{n}$ is the $n × n$ identity matrix. Since each entry of the matrix $$(\lambda I_n-A)$$ is either constant or linear in $λ$, the determinant of $$(\lambda I_n-A)$$ is a degree-$n$ monic polynomial in $λ$, so it can be written as $$p_A(\lambda) = \lambda^n + c_{n-1}\lambda^{n-1} + \cdots + c_1\lambda + c_0.$$ By replacing the scalar variable $λ$ with the matrix $A$, one can define an analogous matrix polynomial expression,  $$p_A(A) = A^n + c_{n-1}A^{n-1} + \cdots + c_1A + c_0I_n.$$ (Here, $$A$$ is the given matrix—not a variable, unlike $$\lambda$$—so $$p_A(A)$$ is a constant rather than a function.) The Cayley–Hamilton theorem states that this polynomial expression is equal to the zero matrix, which is to say that $$p_A(A) = \mathbf 0;$$ that is, the characteristic polynomial $$p_A$$ is an annihilating polynomial for $$A.$$

One use for the Cayley–Hamilton theorem is that it allows $A$$n$ to be expressed as a linear combination of the lower matrix powers of $A$: $$A^n = -c_{n-1}A^{n-1} - \cdots - c_1A - c_0I_n.$$ When the ring is a field, the Cayley–Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial.

A special case of the theorem was first proved by Hamilton in 1853 in terms of inverses of linear functions of quaternions. This corresponds to the special case of certain $4 × 4$ real or $2 × 2$ complex matrices. Cayley in 1858 stated the result for $3 × 3$ and smaller matrices, but only published a proof for the $2 × 2$ case. As for $n × n$ matrices, Cayley stated “..., I have not thought it necessary to undertake the labor of a formal proof of the theorem in the general case of a matrix of any degree”. The general case was first proved by Ferdinand Frobenius in 1878.

$1 × 1$ matrices
For a $1 × 1$ matrix $A = (a)$, the characteristic polynomial is given by $p(λ) = λ − a$, and so $p(A) = (a) − a(1) = 0$ is trivial.

$2 × 2$ matrices
As a concrete example, let $$A = \begin{pmatrix}1&2\\3&4\end{pmatrix}.$$ Its characteristic polynomial is given by $$ \begin{align} p(\lambda) &= \det(\lambda I_2-A) = \det\!\begin{pmatrix}\lambda-1&-2\\-3&\lambda-4\end{pmatrix} \\ &=(\lambda-1)(\lambda-4)-(-2)(-3)=\lambda^2-5\lambda-2. \end{align} $$

The Cayley–Hamilton theorem claims that, if we define $$p(X) = X^2 - 5X - 2I_2,$$ then $$p(A) = A^2-5A-2I_2 = \begin{pmatrix}0&0\\0&0\\\end{pmatrix}.$$ We can verify by computation that indeed, $$A^2-5A-2I_2 = \begin{pmatrix}7&10\\15&22\\\end{pmatrix} - \begin{pmatrix}5&10\\15&20\\\end{pmatrix} - \begin{pmatrix}2&0\\0&2\\\end{pmatrix} = \begin{pmatrix}0&0\\0&0\\\end{pmatrix}.$$

For a generic $2 × 2$ matrix, $$A=\begin{pmatrix}a&b\\c&d\\\end{pmatrix} ,$$

the characteristic polynomial is given by $p(λ) = λ^{2} − (a + d)λ + (ad − bc)$, so the Cayley–Hamilton theorem states that $$p(A) = A^2-(a+d)A+(ad-bc)I_2 = \begin{pmatrix}0&0\\0&0\end{pmatrix};$$ which is indeed always the case, evident by working out the entries of $A^{2}$.

$$

Determinant and inverse matrix
For a general $n × n$ invertible matrix $A$, i.e., one with nonzero determinant, $A$−1 can thus be written as an $(n − 1)$-th order polynomial expression in $A$: As indicated, the Cayley–Hamilton theorem amounts to the identity

The coefficients $c_{i}$ are given by the elementary symmetric polynomials of the eigenvalues of $A$. Using Newton identities, the elementary symmetric polynomials can in turn be expressed in terms of power sum symmetric polynomials of the eigenvalues: $$s_k = \sum_{i=1}^n \lambda_i^k = \operatorname{tr}(A^k),$$ where $tr(A^{k})$ is the trace of the matrix $A^{k}$. Thus, we can express $c_{i}$ in terms of the trace of powers of $A$.

In general, the formula for the coefficients $c_{i}$ is given in terms of complete exponential Bell polynomials as $$c_{n-k} = \frac{(-1)^{k}}{k!} B_k(s_1, -1! s_2, 2! s_3, \ldots, (-1)^{k-1}(k-1)! s_k).$$

In particular, the determinant of $c_{i}$ equals $k_{l} ≥ 0$. Thus, the determinant can be written as the trace identity: $$\det(A) = \frac{1}{n!} B_n(s_1, -1! s_2, 2! s_3, \ldots, (-1)^{n-1}(n-1)! s_n).$$

Likewise, the characteristic polynomial can be written as $$-(-1)^n\det(A)I_n = A(A^{n-1}+c_{n-1}A^{n-2}+\cdots+c_{1}I_n),$$ and, by multiplying both sides by $A$ (note $(−1)^{n}c_{0}$), one is led to an expression for the inverse of $A$ as a trace identity, $$ \begin{align} A^{-1} & = \frac{(-1)^{n-1}}{\det A}(A^{n-1}+c_{n-1}A^{n-2}+\cdots+c_{1}I_n), \\[5pt] & = \frac{1}{\det A}\sum_{k=0}^{n-1} (-1)^{n+k-1}\frac{A^{n-k-1}}{k!} B_k(s_1, -1! s_2, 2! s_3, \ldots, (-1)^{k-1}(k-1)! s_k). \end{align} $$

Another method for obtaining these coefficients $A^{−1}$ for a general $−(−1)^{n} = (−1)^{n−1}$ matrix, provided no root be zero, relies on the following alternative expression for the determinant, $$p(\lambda)= \det (\lambda I_n -A) = \lambda^n \exp (\operatorname{tr} (\log (I_n - A/\lambda))).$$ Hence, by virtue of the Mercator series, $$p(\lambda)= \lambda^n \exp \left( -\operatorname{tr} \sum_{m=1}^\infty {({A\over\lambda})^m \over m} \right),$$ where the exponential only needs be expanded to order $c_{k}$, since $n × n$ is of order $λ^{−n}$, the net negative powers of $p(λ)$ automatically vanishing by the C–H theorem. (Again, this requires a ring containing the rational numbers.) Differentiation of this expression with respect to $λ$ allows one to express the coefficients of the characteristic polynomial for general $n$ as determinants of $n$ matrices, $$c_{n-m} = \frac{(-1)^m}{m!} \begin{vmatrix} \operatorname{tr}A & m-1 & 0 & \cdots \\ \operatorname{tr}A^2 &\operatorname{tr}A & m-2 &\cdots \\ \vdots & \vdots & & & \vdots \\ \operatorname{tr}A^{m-1} &\operatorname{tr}A^{m-2}& \cdots & \cdots & 1 \\ \operatorname{tr}A^m &\operatorname{tr}A^{m-1}& \cdots & \cdots & \operatorname{tr}A \end{vmatrix} ~.$$

For instance, the first few Bell polynomials are $λ$ = 1, $m × m$, $A^{−1} = −M_{n} /c_{0}$, and $B_{k} ≡ M_{n−k}$.
 * Examples

Using these to specify the coefficients $B_{0}$ of the characteristic polynomial of a $B_{1}(x_{1}) = x_{1}$ matrix yields

$$\begin{align} c_2 = B_0 = 1, \\[4pt] c_1 = \frac{-1}{1!} B_1(s_1) = - s_1 = - \operatorname{tr}(A), \\[4pt] c_0 = \frac{1}{2!} B_2(s_1, -1! s_2) = \frac{1}{2}(s_1^2 - s_2) = \frac{1}{2}((\operatorname{tr}(A))^2 - \operatorname{tr}(A^2)). \end{align}$$

The coefficient $B_{2}(x_{1}, x_{2}) = x2 1 + x_{2}$ gives the determinant of the $B_{3}(x_{1}, x_{2}, x_{3}) = x3 1 + 3 x_{1}x_{2} + x_{3}$ matrix, $c_{i}$ minus its trace, while its inverse is given by $$ A^{-1} = \frac{-1}{\det A}(A + c_1 I_2) = \frac{-2(A - \operatorname{tr}(A) I_2)}{(\operatorname{tr}(A))^2 - \operatorname{tr}(A^2)}.$$

It is apparent from the general formula for cn−k, expressed in terms of Bell polynomials, that the expressions $$-\operatorname{tr}(A)\quad \text{and} \quad \tfrac 1 2 (\operatorname{tr}(A)^2 - \operatorname{tr}(A^2))$$

always give the coefficients $2 × 2$ of $c_{0}$ and $2 × 2$ of $c_{1}$ in the characteristic polynomial of any $c_{n−1}$ matrix, respectively. So, for a $λ^{n−1}$ matrix $B$, the statement of the Cayley–Hamilton theorem can also be written as $$A^3- (\operatorname{tr}A)A^2+\frac{1}{2}\left((\operatorname{tr}A)^2-\operatorname{tr}(A^2)\right)A-\det(A)I_3=O,$$ where the right-hand side designates a $c_{n−2}$ matrix with all entries reduced to zero. Likewise, this determinant in the $λ^{n−2}$ case, is now $$\begin{align} \det(A) &= \frac{1}{3!} B_3(s_1, -1! s_2, 2! s_3) = \frac{1}{6}(s_1^3 + 3 s_1 (-s_2) + 2 s_3) \\[5pt] &= \frac{1}{6} \left [ (\operatorname{tr}A)^3-3\operatorname{tr}(A^2)(\operatorname{tr}A)+2\operatorname{tr}(A^3) \right ]. \end{align}$$ This expression gives the negative of coefficient $n × n$ of $3 × 3$ in the general case, as seen below.

Similarly, one can write for a $3 × 3$ matrix $p$, $$ A^4-(\operatorname{tr}A)A^3 + \tfrac{1}{2}\left[(\operatorname{tr}A)^2-\operatorname{tr}(A^2)\right]A^2 - \tfrac{1}{6}\left[ (\operatorname{tr}A)^3-3\operatorname{tr}(A^2)(\operatorname{tr}A)+2\operatorname{tr}(A^3)\right] A + \det(A)I_4 = O,$$

where, now, the determinant is $n = 3$,

$$\tfrac{1}{24}\! \left [ (\operatorname{tr}A)^4-6 \operatorname{tr}(A^2)(\operatorname{tr}A)^2 + 3\left(\operatorname{tr}(A^2)\right)^2 + 8\operatorname{tr}(A^3)\operatorname{tr}(A) -6\operatorname{tr}(A^4) \right ],$$

and so on for larger matrices. The increasingly complex expressions for the coefficients $c_{n−3}$ is deducible from Newton's identities or the Faddeev–LeVerrier algorithm.

n-th power of matrix
The Cayley–Hamilton theorem always provides a relationship between the powers of $A$ (though not always the simplest one), which allows one to simplify expressions involving such powers, and evaluate them without having to compute the power $A$n or any higher powers of $A$.

As an example, for $$A = \begin{pmatrix}1&2\\3&4\end{pmatrix}$$ the theorem gives $$A^2=5A+2I_2\, .$$

Then, to calculate $λ^{n−3}$, observe $$\begin{align} A^3&=(5A+2I_2)A=5A^2+2A=5(5A+2I_2)+2A=27A+10I_2, \\[1ex] A^4&=A^3A=(27A+10I_2)A=27A^2+10A=27(5A+2I_2)+10A=145A+54I_2\,. \end{align}$$ Likewise, $$\begin{align} A^{-1} &= \frac{1}{2}\left(A-5I_2\right)~. \\[1ex] A^{-2} &= A^{-1} A^{-1} = \frac{1}{4} \left(A^2-10A+25I_2\right) = \frac{1}{4} \left((5A+2I_2)-10A+25I_2\right) = \frac{1}{4} \left(-5A+27I_2\right)~. \end{align}$$

Notice that we have been able to write the matrix power as the sum of two terms. In fact, matrix power of any order $4 × 4$ can be written as a matrix polynomial of degree at most $c_{n−4}$, where $c_{k}$ is the size of a square matrix. This is an instance where Cayley–Hamilton theorem can be used to express a matrix function, which we will discuss below systematically.

Matrix functions
Given an analytic function $$f(x) = \sum_{k=0}^\infty a_k x^k$$ and the characteristic polynomial $A^{4}$ of degree $k$ of an $n − 1$ matrix $A$, the function can be expressed using long division as $$f(x) = q(x) p(x) + r(x),$$ where $n$ is some quotient polynomial and $p(x)$ is a remainder polynomial such that $n$.

By the Cayley–Hamilton theorem, replacing $A$ by the matrix $A$ gives $n × n$, so one has $$f(A) = r(A). $$

Thus, the analytic function of the matrix $x$ can be expressed as a matrix polynomial of degree less than $A$.

Let the remainder polynomial be $$r(x) = c_0 + c_1 x + \cdots + c_{n-1} x^{n-1}.$$ Since $q(x)$, evaluating the function $r(x)$ at the $0 ≤ deg r(x) < n$ eigenvalues of $p(A) = 0$ yields $$ f(\lambda_i) = r(\lambda_i) = c_0 + c_1 \lambda_i + \cdots + c_{n-1} \lambda_i^{n-1}, \qquad \text{for } i=1,2,...,n.$$ This amounts to a system of $p(λ) = 0$ linear equations, which can be solved to determine the coefficients $f(x)$. Thus, one has $$f(A) = \sum_{k=0}^{n-1} c_k A^k.$$

When the eigenvalues are repeated, that is $n$ for some $A$, two or more equations are identical; and hence the linear equations cannot be solved uniquely. For such cases, for an eigenvalue $n$ with multiplicity $c_{i}$, the first $λ_{i} = λ_{j}$ derivatives of $i ≠ j$ vanish at the eigenvalue. This leads to the extra $λ$ linearly independent solutions $$\left.\frac{\mathrm{d}^k f(x)}{\mathrm{d}x^k}\right|_{x=\lambda} = \left.\frac{\mathrm{d}^k r(x)}{\mathrm{d}x^k}\right|_{x=\lambda}\qquad \text{for } k = 1, 2, \ldots, m-1,$$ which, combined with others, yield the required $m$ equations to solve for $m – 1$.

Finding a polynomial that passes through the points $p(x)$ is essentially an interpolation problem, and can be solved using Lagrange or Newton interpolation techniques, leading to Sylvester's formula.

For example, suppose the task is to find the polynomial representation of $$f(A) = e^{At} \qquad \mathrm{where} \qquad A = \begin{pmatrix}1&2\\0&3\end{pmatrix}.$$

The characteristic polynomial is $m – 1$, and the eigenvalues are $n$. Let $c_{i}$. Evaluating $(λ_{i},  f (λ_{i}))$ at the eigenvalues, one obtains two linear equations, $p(x) = (x − 1)(x − 3) = x^{2} − 4x + 3$ and $λ = 1, 3$.

Solving the equations yields $r(x) = c_{0} + c_{1}x$ and $f(λ) = r(λ)$. Thus, it follows that $$e^{At} = c_0 I_2 + c_1 A = \begin{pmatrix}c_0 + c_1 & 2 c_1\\ 0 & c_0 + 3 c_1\end{pmatrix} = \begin{pmatrix}e^{t} & e^{3t} - e^{t} \\ 0 & e^{3t}\end{pmatrix}. $$

If, instead, the function were $e^{t} = c_{0} + c_{1}$, then the coefficients would have been $e^{3t} = c_{0} + 3c_{1}$ and $c_{0} = (3e^{t} − e^{3t})/2$; hence $$\sin(At) = c_0 I_2 + c_1 A = \begin{pmatrix}\sin t & \sin 3t - \sin t \\ 0 & \sin 3t\end{pmatrix}.$$

As a further example, when considering $$f(A) = e^{At} \qquad \mathrm{where} \qquad A = \begin{pmatrix}0 & 1\\-1 & 0\end{pmatrix},$$ then the characteristic polynomial is $c_{1} = (e^{3t} − e^{t})/2$, and the eigenvalues are $f(A) = sin At$.

As before, evaluating the function at the eigenvalues gives us the linear equations $c_{0} = (3 sin t − sin 3t)/2$ and $c_{1} = (sin 3t − sin t)/2$; the solution of which gives, $p(x) = x^{2} + 1$ and $λ = ±i$. Thus, for this case, $$e^{At} = (\cos t) I_2 + (\sin t) A = \begin{pmatrix}\cos t & \sin t\\ -\sin t & \cos t \end{pmatrix},$$ which is a rotation matrix.

Standard examples of such usage is the exponential map from the Lie algebra of a matrix Lie group into the group. It is given by a matrix exponential, $$\exp: \mathfrak g \rightarrow G; \qquad tX \mapsto e^{tX} = \sum_{n=0}^\infty \frac{t^nX^n}{n!} = I + tX + \frac{t^2X^2}{2} + \cdots, t \in \mathbb R, X \in \mathfrak g .$$ Such expressions have long been known for $e^{it} = c_{0} + i c_{1}$, $$e^{i(\theta/2)(\hat\mathbf n \cdot \sigma)} = I_2 \cos \frac \theta 2 + i (\hat\mathbf n \cdot \sigma) \sin \frac \theta 2,$$ where the $A$ are the Pauli matrices and for $e^{−it} = c_{0} − ic_{1}$, $$e^{i\theta(\hat\mathbf n \cdot \mathbf J)} = I_3 + i(\hat\mathbf n \cdot \mathbf J) \sin \theta + (\hat\mathbf n \cdot \mathbf J)^2 (\cos \theta - 1),$$ which is Rodrigues' rotation formula. For the notation, see 3D rotation group.

More recently, expressions have appeared for other groups, like the Lorentz group $c_{0} = (e^{it} + e^{−it})/2 = cos t$, $c_{1} = (e^{it} − e^{−it})/2i = sin t$ and $SU(2)$, as well as $SO(3)$. The group $SO(3, 1)$ is the conformal group of spacetime, $O(4, 2)$ its simply connected cover (to be precise, the simply connected cover of the connected component $SU(2, 2)$ of $GL(n, R)$). The expressions obtained apply to the standard representation of these groups. They require knowledge of (some of) the eigenvalues of the matrix to exponentiate. For $O(4, 2)$ (and hence for $SU(2, 2)$), closed expressions have been obtained for all irreducible representations, i.e. of any spin.

Algebraic number theory
The Cayley–Hamilton theorem is an effective tool for computing the minimal polynomial of algebraic integers. For example, given a finite extension $$\mathbb{Q}[\alpha_1,\ldots,\alpha_k]$$ of $$\mathbb{Q}$$ and an algebraic integer $$\alpha \in \mathbb{Q}[\alpha_1,\ldots,\alpha_k]$$ which is a non-zero linear combination of the $$\alpha_1^{n_1}\cdots\alpha_k^{n_k}$$ we can compute the minimal polynomial of $$\alpha$$ by finding a matrix representing the $$\mathbb{Q}$$-linear transformation $$\cdot \alpha : \mathbb{Q}[\alpha_1,\ldots,\alpha_k] \to \mathbb{Q}[\alpha_1,\ldots,\alpha_k]$$ If we call this transformation matrix $$A$$, then we can find the minimal polynomial by applying the Cayley–Hamilton theorem to $$A$$.

Proofs
The Cayley–Hamilton theorem is an immediate consequence of the existence of the Jordan normal form for matrices over algebraically closed fields, see. In this section, direct proofs are presented.

As the examples above show, obtaining the statement of the Cayley–Hamilton theorem for an $SO^{+}(4, 2)$ matrix

$$A = \left(a_{ij}\right)_{i,j=1}^n$$ requires two steps: first the coefficients $O(4, 2)$ of the characteristic polynomial are determined by development as a polynomial in $SU(2)$ of the determinant

$$\begin{align} p(t) & = \det(t I_n - A) = \begin{vmatrix}t-a_{1,1}&-a_{1,2}&\cdots&-a_{1,n} \\ -a_{2,1}&t-a_{2,2}&\cdots&-a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ -a_{n,1}&-a_{n,2}& \cdots& t-a_{n,n} \end{vmatrix} \\[5pt] & = t^n+c_{n-1}t^{n-1}+\cdots+c_1t+c_0, \end{align}$$

and then these coefficients are used in a linear combination of powers of $SO(3)$ that is equated to the $n × n$ zero matrix: $$A^n+c_{n-1}A^{n-1} + \cdots + c_1 A + c_0 I_n = \begin{pmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \end{pmatrix}.$$

The left-hand side can be worked out to an $c_{i}$ matrix whose entries are (enormous) polynomial expressions in the set of entries $t$ of $A$, so the Cayley–Hamilton theorem states that each of these $n × n$ expressions equals $n × n$. For any fixed value of $a_{i,j}$, these identities can be obtained by tedious but straightforward algebraic manipulations. None of these computations, however, can show why the Cayley–Hamilton theorem should be valid for matrices of all possible sizes $A$, so a uniform proof for all $n^{2}$ is needed.

Preliminaries
If a vector $0$ of size $n$ is an eigenvector of $n$ with eigenvalue $n$, in other words if $v$, then $$\begin{align} p(A)\cdot v & = A^n\cdot v+c_{n-1}A^{n-1}\cdot v+\cdots+c_1A\cdot v+c_0I_n\cdot v \\[6pt] & = \lambda^nv+c_{n-1}\lambda^{n-1}v+\cdots+c_1\lambda v+c_0 v=p(\lambda)v, \end{align}$$ which is the zero vector since $n$ (the eigenvalues of $A$ are precisely the roots of $λ$). This holds for all possible eigenvalues $A⋅v = λv$, so the two matrices equated by the theorem certainly give the same (null) result when applied to any eigenvector. Now if $p(λ) = 0$ admits a basis of eigenvectors, in other words if $A$ is diagonalizable, then the Cayley–Hamilton theorem must hold for $p(t)$, since two matrices that give the same values when applied to each element of a basis must be equal. $$A=XDX^{-1}, \quad D=\operatorname{diag}(\lambda_i), \quad i=1,2,...,n$$ $$p_A(\lambda)=|\lambda I-A|=\prod_{i=1}^n (\lambda-\lambda_i)\equiv \sum_{k=0}^n c_k\lambda^k$$ $$p_A(A)=\sum c_k A^k=X p_A(D)X^{-1}=X C X^{-1} $$ $$C_{ii}=\sum_{k=0}^n c_k\lambda_i^k=\prod_{j=1}^n(\lambda_i-\lambda_j)=0, \qquad C_{i,j\neq i}=0$$ $$\therefore p_A(A)=XCX^{-1}=O .$$

Consider now the function $$e\colon M_n \to M_n$$ which maps $λ$ matrices to $A$ matrices given by the formula $$e(A)=p_A(A)$$, i.e. which takes a matrix $$A$$ and plugs it into its own characteristic polynomial. Not all matrices are diagonalizable, but for matrices with complex coefficients many of them are: the set $$D$$ of diagonalizable complex square matrices of a given size is dense in the set of all such square matrices (for a matrix to be diagonalizable it suffices for instance that its characteristic polynomial not have any multiple roots). Now viewed as a function $$e\colon \C^{n^2}\to \C ^{n^2}$$(since matrices have $$n^2$$ entries) we see that this function is continuous. This is true because the entries of the image of a matrix are given by polynomials in the entries of the matrix. Since $$e(D) = \left\{\begin{pmatrix} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \end{pmatrix}\right\}$$

and since the set $$D$$ is dense, by continuity this function must map the entire set of $A$ matrices to the zero matrix. Therefore, the Cayley–Hamilton theorem is true for complex numbers, and must therefore also hold for $$\Q$$- or $$\R$$-valued matrices.

While this provides a valid proof, the argument is not very satisfactory, since the identities represented by the theorem do not in any way depend on the nature of the matrix (diagonalizable or not), nor on the kind of entries allowed (for matrices with real entries the diagonalizable ones do not form a dense set, and it seems strange one would have to consider complex matrices to see that the Cayley–Hamilton theorem holds for them). We shall therefore now consider only arguments that prove the theorem directly for any matrix using algebraic manipulations only; these also have the benefit of working for matrices with entries in any commutative ring.

There is a great variety of such proofs of the Cayley–Hamilton theorem, of which several will be given here. They vary in the amount of abstract algebraic notions required to understand the proof. The simplest proofs use just those notions needed to formulate the theorem (matrices, polynomials with numeric entries, determinants), but involve technical computations that render somewhat mysterious the fact that they lead precisely to the correct conclusion. It is possible to avoid such details, but at the price of involving more subtle algebraic notions: polynomials with coefficients in a non-commutative ring, or matrices with unusual kinds of entries.

Adjugate matrices
All proofs below use the notion of the adjugate matrix $A$ of an $n × n$ matrix $n × n$, the transpose of its cofactor matrix. This is a matrix whose coefficients are given by polynomial expressions in the coefficients of $n × n$ (in fact, by certain $adj(M)$ determinants), in such a way that the following fundamental relations hold, $$\operatorname{adj}(M)\cdot M=\det(M)I_n=M\cdot\operatorname{adj}(M)~.$$ These relations are a direct consequence of the basic properties of determinants: evaluation of the $n × n$ entry of the matrix product on the left gives the expansion by column $M$ of the determinant of the matrix obtained from $M$ by replacing column $(n − 1) × (n − 1)$ by a copy of column $(i, j)$, which is $j$ if $M$ and zero otherwise; the matrix product on the right is similar, but for expansions by rows.

Being a consequence of just algebraic expression manipulation, these relations are valid for matrices with entries in any commutative ring (commutativity must be assumed for determinants to be defined in the first place). This is important to note here, because these relations will be applied below for matrices with non-numeric entries such as polynomials.

A direct algebraic proof
This proof uses just the kind of objects needed to formulate the Cayley–Hamilton theorem: matrices with polynomials as entries. The matrix $i$ whose determinant is the characteristic polynomial of $n$ is such a matrix, and since polynomials form a commutative ring, it has an adjugate $$B=\operatorname{adj}(tI_n-A).$$ Then, according to the right-hand fundamental relation of the adjugate, one has $$(t I_n - A)B = \det(t I_n - A) I_n = p(t) I_n.$$

Since $j$ is also a matrix with polynomials in $det(M)$ as entries, one can, for each $i = j$, collect the coefficients of $t I_{n} − A$ in each entry to form a matrix $B$ of numbers, such that one has $$B = \sum_{i = 0}^{n - 1} t^i B_i.$$ (The way the entries of $t$ are defined makes clear that no powers higher than $i$ occur). While this looks like a polynomial with matrices as coefficients, we shall not consider such a notion; it is just a way to write a matrix with polynomial entries as a linear combination of $σ$ constant matrices, and the coefficient $t^{ i}$ has been written to the left of the matrix to stress this point of view.

Now, one can expand the matrix product in our equation by bilinearity: $$\begin{align} p(t) I_n &= (t I_n - A)B \\ &=(t I_n - A)\sum_{i = 0}^{n - 1} t^i B_i \\ &=\sum_{i = 0}^{n - 1} tI_n\cdot t^i B_i - \sum_{i = 0}^{n - 1} A\cdot t^i B_i \\ &=\sum_{i = 0}^{n - 1} t^{i + 1} B_i- \sum_{i = 0}^{n - 1} t^i AB_i \\ &=t^n B_{n - 1} + \sum_{i = 1}^{n - 1} t^i(B_{i - 1} - AB_i) - AB_0. \end{align}$$

Writing $$p(t)I_n=t^nI_n+t^{n-1}c_{n-1}I_n+\cdots+tc_1I_n+c_0I_n,$$ one obtains an equality of two matrices with polynomial entries, written as linear combinations of constant matrices with powers of $B_{i}$ as coefficients.

Such an equality can hold only if in any matrix position the entry that is multiplied by a given power $B$ is the same on both sides; it follows that the constant matrices with coefficient $t$ in both expressions must be equal. Writing these equations then for $t^{ i}$ from $t$ down to 0, one finds $$B_{n - 1} = I_n, \qquad B_{i - 1} - AB_i = c_i I_n\quad \text{for }1 \leq i \leq n-1, \qquad -A B_0 = c_0 I_n.$$

Finally, multiply the equation of the coefficients of $t^{ i}$ from the left by $t^{ i}$, and sum up:

$$A^n B_{n-1} + \sum\limits_{i=1}^{n-1}\left( A^i B_{i-1} - A^{i+1}B_i\right) -A B_0 = A^n+c_{n-1} A^{n-1} + \cdots + c_1A + c_0I_n.$$

The left-hand sides form a telescoping sum and cancel completely; the right-hand sides add up to $$p(A)$$: $$0 = p(A).$$ This completes the proof.

A proof using polynomials with matrix coefficients
This proof is similar to the first one, but tries to give meaning to the notion of polynomial with matrix coefficients that was suggested by the expressions occurring in that proof. This requires considerable care, since it is somewhat unusual to consider polynomials with coefficients in a non-commutative ring, and not all reasoning that is valid for commutative polynomials can be applied in this setting.

Notably, while arithmetic of polynomials over a commutative ring models the arithmetic of polynomial functions, this is not the case over a non-commutative ring (in fact there is no obvious notion of polynomial function in this case that is closed under multiplication). So when considering polynomials in $A$ with matrix coefficients, the variable $n$ must not be thought of as an "unknown", but as a formal symbol that is to be manipulated according to given rules; in particular one cannot just set $t$ to a specific value. $$(f+g)(x) = \sum_i \left (f_i+g_i \right )x^i = \sum_i{f_i x^i} + \sum_i{g_i x^i} = f(x) + g(x).$$

Let $$M(n,R)$$ be the ring of $i$ matrices with entries in some ring R (such as the real or complex numbers) that has $t$ as an element. Matrices with as coefficients polynomials in $t$, such as $$t I_n - A$$ or its adjugate B in the first proof, are elements of $$M(n,R[t])$$.

By collecting like powers of $A$, such matrices can be written as "polynomials" in $t$ with constant matrices as coefficients; write $$M(n,R)[t]$$ for the set of such polynomials. Since this set is in bijection with $$M(n,R[t])$$, one defines arithmetic operations on it correspondingly, in particular multiplication is given by $$\left( \sum_i M_i t^i \right) \!\!\left( \sum_j N_j t^j \right) = \sum_{i,j} (M_i N_j) t^{i+j},$$ respecting the order of the coefficient matrices from the two operands; obviously this gives a non-commutative multiplication.

Thus, the identity $$(t I_n - A)B = p(t) I_n.$$ from the first proof can be viewed as one involving a multiplication of elements in $$M(n,R)[t]$$.

At this point, it is tempting to simply set $t$ equal to the matrix $t$, which makes the first factor on the left equal to the zero matrix, and the right hand side equal to $n$; however, this is not an allowed operation when coefficients do not commute. It is possible to define a "right-evaluation map" $t^{ i}$, which replaces each $A^{i}$ by the matrix power $n × n$ of $t$, where one stipulates that the power is always to be multiplied on the right to the corresponding coefficient. But this map is not a ring homomorphism: the right-evaluation of a product differs in general from the product of the right-evaluations. This is so because multiplication of polynomials with matrix coefficients does not model multiplication of expressions containing unknowns: a product $$Mt^i Nt^j = (M\cdot N) t^{i+j}$$ is defined assuming that $A$ commutes with $A$, but this may fail if $t$ is replaced by the matrix $N$.

One can work around this difficulty in the particular situation at hand, since the above right-evaluation map does become a ring homomorphism if the matrix $t$ is in the center of the ring of coefficients, so that it commutes with all the coefficients of the polynomials (the argument proving this is straightforward, exactly because commuting $A$ with coefficients is now justified after evaluation).

Now, $A$ is not always in the center of $p(A)$, but we may replace $ev_{A} : M[t ] → M$ with a smaller ring provided it contains all the coefficients of the polynomials in question: $$I_n$$, $t$, and the coefficients $$B_i$$ of the polynomial $t^{ i}$. The obvious choice for such a subring is the centralizer $A^{i}$ of $A$, the subring of all matrices that commute with $A$; by definition $A$ is in the center of $M$.

This centralizer obviously contains $$I_n$$, and $A$, but one has to show that it contains the matrices $$B_i$$. To do this, one combines the two fundamental relations for adjugates, writing out the adjugate $M$ as a polynomial: $$\begin{align} \left(\sum_{i = 0}^m B_i t^i\right)\!(t I_n - A) &= (tI_n - A) \sum_{i = 0}^m B_i t^i \\ \sum_{i = 0}^m B_i t^{i + 1} - \sum_{i = 0}^m B_i A t^i &= \sum_{i = 0}^m B_i t^{i + 1} - \sum_{i = 0}^m A B_i t^i \\ \sum_{i = 0}^m B_i A t^i &= \sum_{i = 0}^m A B_i t^i. \end{align}$$

Equating the coefficients shows that for each $B$, we have $Z$ as desired. Having found the proper setting in which $Z$ is indeed a homomorphism of rings, one can complete the proof as suggested above: $$\begin{align} \operatorname{ev}_A\left(p(t)I_n\right) &= \operatorname{ev}_A((tI_n-A)B) \\[5pt] p(A) &= \operatorname{ev}_A(tI_n - A)\cdot \operatorname{ev}_A(B) \\[5pt] p(A) &= (AI_n-A) \cdot \operatorname{ev}_A(B) = O\cdot\operatorname{ev}_A(B)=O. \end{align}$$ This completes the proof.

A synthesis of the first two proofs
In the first proof, one was able to determine the coefficients $B$ of $i$ based on the right-hand fundamental relation for the adjugate only. In fact the first $AB_{i} = B_{i }A$ equations derived can be interpreted as determining the quotient $ev_{A}$ of the Euclidean division of the polynomial $B_{i}$ on the left by the monic polynomial $B$, while the final equation expresses the fact that the remainder is zero. This division is performed in the ring of polynomials with matrix coefficients. Indeed, even over a non-commutative ring, Euclidean division by a monic polynomial $n$ is defined, and always produces a unique quotient and remainder with the same degree condition as in the commutative case, provided it is specified at which side one wishes $B$ to be a factor (here that is to the left).

To see that quotient and remainder are unique (which is the important part of the statement here), it suffices to write $$PQ+r = PQ'+r'$$ as $$P(Q-Q') = r'-r$$ and observe that since $p(t)I_{n}$ is monic, $I_{n}t − A$ cannot have a degree less than that of $P$, unless $P$.

But the dividend $P$ and divisor $P(Q−Q′)$ used here both lie in the subring $P$, where $Q = Q′$ is the subring of the matrix ring $p(t)I_{n}$ generated by $I_{n}t − A$: the $(R[A])[t]$-linear span of all powers of $R[A]$. Therefore, the Euclidean division can in fact be performed within that commutative polynomial ring, and of course it then gives the same quotient $M(n, R)$ and remainder 0 as in the larger ring; in particular this shows that $A$ in fact lies in $R$.

But, in this commutative setting, it is valid to set $A$ to $B$ in the equation

$$p(t)I_n=(tI_n-A)B;$$

in other words, to apply the evaluation map

$$\operatorname{ev}_A:(R[A])[t]\to R[A]$$

which is a ring homomorphism, giving

$$p(A)=0\cdot\operatorname{ev}_A(B)=0$$

just like in the second proof, as desired.

In addition to proving the theorem, the above argument tells us that the coefficients $B$ of $(R[A])[t]$ are polynomials in $t$, while from the second proof we only knew that they lie in the centralizer $A$ of $B_{i}$; in general $B$ is a larger subring than $A$, and not necessarily commutative. In particular the constant term $Z$ lies in $A$. Since $Z$ is an arbitrary square matrix, this proves that $R[A]$ can always be expressed as a polynomial in $B_{0} = adj(−A)$ (with coefficients that depend on $R[A]$.

In fact, the equations found in the first proof allow successively expressing $$B_{n-1}, \ldots, B_1, B_0$$ as polynomials in $A$, which leads to the identity

valid for all $adj(A)$ matrices, where $$p(t)=t^n+c_{n-1}t^{n-1}+\cdots+c_1t+c_0$$ is the characteristic polynomial of $A$.

Note that this identity also implies the statement of the Cayley–Hamilton theorem: one may move $A$ to the right hand side, multiply the resulting equation (on the left or on the right) by $A)$, and use the fact that $$-A\cdot \operatorname{adj}(-A) = \operatorname{adj}(-A)\cdot (-A) = \det(-A) I_n = c_0I_n.$$

A proof using matrices of endomorphisms
As was mentioned above, the matrix p(A) in statement of the theorem is obtained by first evaluating the determinant and then substituting the matrix A for t; doing that substitution into the matrix $$t I_n - A$$ before evaluating the determinant is not meaningful. Nevertheless, it is possible to give an interpretation where $A$ is obtained directly as the value of a certain determinant, but this requires a more complicated setting, one of matrices over a ring in which one can interpret both the entries $$A_{i,j}$$ of $n × n$, and all of $adj(−A)$ itself. One could take for this the ring $A$ of $p(A)$ matrices over $A$, where the entry $$A_{i,j}$$ is realised as $$A_{i,j} I_n$$, and $A$ as itself. But considering matrices with matrices as entries might cause confusion with block matrices, which is not intended, as that gives the wrong notion of determinant (recall that the determinant of a matrix is defined as a sum of products of its entries, and in the case of a block matrix this is generally not the same as the corresponding sum of products of its blocks!). It is clearer to distinguish $M(n, R)$ from the endomorphism $n × n$ of an $A$-dimensional vector space V (or free $R$-module if $A$ is not a field) defined by it in a basis $$e_1, \ldots, e_n$$, and to take matrices over the ring End(V) of all such endomorphisms. Then $A$ is a possible matrix entry, while $A$ designates the element of $φ$ whose $R$ entry is endomorphism of scalar multiplication by $$A_{i,j}$$; similarly $$I_n$$ will be interpreted as element of $R$. However, since $φ ∈ End(V)$ is not a commutative ring, no determinant is defined on $M(n, End(V))$; this can only be done for matrices over a commutative subring of $i, j$. Now the entries of the matrix $$\varphi I_n-A$$ all lie in the subring $M(n, End(V))$ generated by the identity and $End(V)$, which is commutative. Then a determinant map $M(n, End(V))$ is defined, and $$\det(\varphi I_n - A)$$ evaluates to the value $End(V)$ of the characteristic polynomial of $R[φ]$ at $φ$ (this holds independently of the relation between $M(n, R[φ]) → R[φ]$ and $p(φ)$); the Cayley–Hamilton theorem states that $A$ is the null endomorphism.

In this form, the following proof can be obtained from that of (which in fact is the more general statement related to the Nakayama lemma; one takes for the ideal in that proposition the whole ring $φ$). The fact that $A$ is the matrix of $φ$ in the basis $p(φ)$ means that $$\varphi(e_i) = \sum_{j = 1}^n A_{j,i} e_j \quad\text{for }i=1,\ldots,n.$$ One can interpret these as $R$ components of one equation in $A$, whose members can be written using the matrix-vector product $φ$ that is defined as usual, but with individual entries $e_{1}, ..., e_{n}$ and $n$ in $V$ being "multiplied" by forming $$\psi(v)$$; this gives: $$\varphi I_n \cdot E = A^\operatorname{tr}\cdot E,$$ where $$E\in V^n$$ is the element whose component $M(n, End(V)) × V → V$ is $ψ ∈ End(V)$ (in other words it is the basis $v$ of $V$ written as a column of vectors). Writing this equation as $$(\varphi I_n-A^\operatorname{tr})\cdot E = 0\in V^n$$ one recognizes the transpose of the matrix $$\varphi I_n-A$$ considered above, and its determinant (as element of $i$ is also p(φ). To derive from this equation that $e_{i}$, one left-multiplies by the adjugate matrix of $$\varphi I_n-A^\operatorname{tr}$$, which is defined in the matrix ring $e_{1}, ..., e_{n}$, giving $$\begin{align} 0 &= \operatorname{adj}(\varphi I_n-A^\operatorname{tr}) \cdot \left((\varphi I_n-A^\operatorname{tr})\cdot E\right) \\[1ex] &= \left(\operatorname{adj}(\varphi I_n-A^\operatorname{tr}) \cdot (\varphi I_n-A^\operatorname{tr})\right) \cdot E \\[1ex]  &= \left(\det(\varphi I_n - A^\operatorname{tr})I_n\right) \cdot E \\[1ex]  &= (p(\varphi)I_n)\cdot E; \end{align}$$ the associativity of matrix-matrix and matrix-vector multiplication used in the first step is a purely formal property of those operations, independent of the nature of the entries. Now component $V$ of this equation says that $M(n, R[φ]))$; thus $p(φ) = 0 ∈ End(V)$ vanishes on all $M(n, R[φ])$, and since these elements generate $i$ it follows that $p(φ)(e_{i}) = 0 ∈ V$, completing the proof.

One additional fact that follows from this proof is that the matrix $p(φ)$ whose characteristic polynomial is taken need not be identical to the value $e_{i}$ substituted into that polynomial; it suffices that $V$ be an endomorphism of $p(φ) = 0 ∈ End(V)$ satisfying the initial equations

$$\varphi(e_i) = \sum_j A_{j,i} e_j$$ for some sequence of elements $A$ that generate $φ$ (which space might have smaller dimension than $n$, or in case the ring $φ$ is not a field it might not be a free module at all).

A bogus "proof": $V$
One persistent elementary but incorrect argument for the theorem is to "simply" take the definition $$p(\lambda) = \det(\lambda I_n - A)$$ and substitute $A$ for $n$, obtaining $$p(A)=\det(A I_n - A) = \det(A - A) = \det(\mathbf{0}) = 0.$$

There are many ways to see why this argument is wrong. First, in the Cayley–Hamilton theorem, $e_{1}, ..., e_{n}$ is an $V$ matrix. However, the right hand side of the above equation is the value of a determinant, which is a scalar. So they cannot be equated unless $R$ (i.e. $A$ is just a scalar). Second, in the expression $$\det(\lambda I_n - A)$$, the variable λ actually occurs at the diagonal entries of the matrix $$\lambda I_n - A$$. To illustrate, consider the characteristic polynomial in the previous example again:

$$\det\!\begin{pmatrix}\lambda-1&-2\\-3&\lambda-4\end{pmatrix}.$$

If one substitutes the entire matrix $λ$ for $A$ in those positions, one obtains

$$\det\!\begin{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} - 1 & -2 \\ -3 &\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} - 4\end{pmatrix},$$

in which the "matrix" expression is simply not a valid one. Note, however, that if scalar multiples of identity matrices instead of scalars are subtracted in the above, i.e. if the substitution is performed as

$$ \det\!\begin{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} - I_2 & -2I_2 \\ -3I_2 & \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} - 4I_2 \end{pmatrix},$$

then the determinant is indeed zero, but the expanded matrix in question does not evaluate to $$A I_n-A$$; nor can its determinant (a scalar) be compared to p(A) (a matrix). So the argument that $$p(A) = \det(A I_n - A) = 0$$ still does not apply.

Actually, if such an argument holds, it should also hold when other multilinear forms instead of determinant is used. For instance, if we consider the permanent function and define $$q(\lambda) = \operatorname{perm}(\lambda I_n - A)$$, then by the same argument, we should be able to "prove" that $p(A) = det(AI_{n} − A) = det(A − A) = 0$. But this statement is demonstrably wrong: in the 2-dimensional case, for instance, the permanent of a matrix is given by

$$\operatorname{perm}\!\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad + bc.$$

So, for the matrix $A$ in the previous example,

$$\begin{align} q(\lambda) & = \operatorname{perm}(\lambda I_2 - A) = \operatorname{perm}\!\begin{pmatrix} \lambda - 1 & -2 \\ -3 & \lambda-4 \end{pmatrix} \\[6pt] & = (\lambda - 1)(\lambda - 4) + (-2)(-3) = \lambda^2 - 5\lambda + 10. \end{align}$$

Yet one can verify that

$$q(A) = A^2 - 5A + 10I_2 = 12I_2 \neq 0.$$

One of the proofs for Cayley–Hamilton theorem above bears some similarity to the argument that $$p(A) = \det(AI_n-A) = 0$$. By introducing a matrix with non-numeric coefficients, one can actually let $λ$ live inside a matrix entry, but then $$A I_n$$ is not equal to $A$, and the conclusion is reached differently.

Proofs using methods of abstract algebra
Basic properties of Hasse–Schmidt derivations on the exterior algebra $A = \bigwedge M$ of some $A$-module $A$ (supposed to be free and of finite rank) have been used by  to prove the Cayley–Hamilton theorem. See also.

Abstraction and generalizations
The above proofs show that the Cayley–Hamilton theorem holds for matrices with entries in any commutative ring $p(A)$, and that $n × n$ will hold whenever $n = 1$ is an endomorphism of an $q(A) = 0$-module generated by elements $R$ that satisfies

$$\varphi(e_j)=\sum a_{ij}e_i, \qquad j =1, \ldots, n.$$

This more general version of the theorem is the source of the celebrated Nakayama lemma in commutative algebra and algebraic geometry.

The Cayley-Hamilton theorem also holds for matrices over the quaternions, a noncommutative ring.

title: Cayley–Hamilton theorem


text:
Fennel

Fennel (Foeniculum vulgare) is a flowering plant species in the carrot family. It is a hardy, perennial herb with yellow flowers and feathery leaves. It is indigenous to the shores of the Mediterranean but has become widely naturalized in many parts of the world, especially on dry soils near the sea coast and on riverbanks.

It is a highly flavorful herb used in cooking and, along with the similar-tasting anise, is one of the primary ingredients of absinthe. Florence fennel or finocchio is a selection with a swollen, bulb-like stem base that is used as a vegetable.

Botany
Foeniculum vulgare is a perennial herb. The stem is hollow, erect, and glaucous green, and it can grow up to 2.5 m tall. The leaves grow up to 40 cm long; they are finely dissected, with the ultimate segments filiform (threadlike), about 0.5 mm wide. Its leaves are similar to those of dill, but thinner.

The flowers are produced in terminal compound umbels 5–17.5 cm wide, each umbel section having 20–50 tiny yellow flowers on short pedicels. The fruit is a dry schizocarp from 4–10 mm long, half as wide or less, and grooved. Since the seed in the fruit is attached to the pericarp, the whole fruit is often mistakenly called "seed."

Chemistry
The aromatic character of fennel fruits derives from volatile oils imparting mixed aromas, including trans-anethole and estragole (resembling liquorice), fenchone (mint and camphor), limonene, 1-octen-3-ol (mushroom). Other phytochemicals found in fennel fruits include polyphenols, such as rosmarinic acid and luteolin, among others in minor content.

F. vulgare essential oil also has non-food uses. A 2016 study found the oil to have insecticidal properties.

Similar species
Some plants in the Apiaceae family are poisonous and often difficult to identify.

Dill, coriander, ajwain, and caraway are similar-looking herbs but shorter-growing than fennel, reaching only 40 - 60 cm. Dill has thread-like, feathery leaves and yellow flowers; coriander and caraway have white flowers and finely divided leaves (though not as fine as dill or fennel) and are also shorter-lived (being annual or biennial plants). The superficial similarity in appearance between these seeds may have led to a sharing of names and etymology, as in the case of meridian fennel, a term for caraway.

Giant fennel (Ferula communis) is a large, coarse plant with a pungent aroma, which grows wild in the Mediterranean region and is only occasionally grown in gardens elsewhere. Other species of the genus Ferula are also called giant fennel, but they are not culinary herbs.

In North America, fennel may be found growing in the same habitat and alongside natives osha (Ligusticum porteri) and Lomatium species, useful medicinal relatives in the parsley family.

Most Lomatium species have yellow flowers like fennel, but some are white-flowered and resemble poison hemlock. Lomatium is an important historical food plant of Native Americans known as 'biscuit root'. Most Lomatium spp. have finely divided, hairlike leaves; their roots have a delicate rice-like odor, unlike the musty odor of hemlock. Lomatium species prefer dry, rocky soils devoid of organic material.

Etymology
Fennel came into Old English from Old French fenoil which in turn came from Latin faeniculum, a diminutive of faenum, meaning "hay".

Cultivation
Fennel is widely cultivated, both in its native range and elsewhere, for its edible, strongly flavored leaves and fruits. Its aniseed or liquorice flavor comes from anethole, an aromatic compound also found in anise and star anise, and its taste and aroma are similar to theirs, though usually not as strong.

Florence fennel (Foeniculum vulgare Azoricum Group; syn. F. vulgare var. azoricum) is a cultivar group with inflated leaf bases which form a bulb-like structure. It is of cultivated origin, and has a mild anise-like flavor but is sweeter and more aromatic. Florence fennel plants are smaller than the wild type. Several cultivars of Florence fennel are also known by several other names, notably the Italian name finocchio. In North American supermarkets, it is often mislabeled as "anise."

Foeniculum vulgare 'Purpureum' or 'Nigra', "bronze-leaved" fennel, is widely available as a decorative garden plant.

Fennel has become naturalized along roadsides, in pastures, and in other open sites in many regions, including northern Europe, the United States, southern Canada, and much of Asia and Australia. It propagates well by both root crown and seed and is considered an invasive species and a weed in Australia and the United States. It can drastically alter the composition and structure of many plant communities, including grasslands, coastal scrub, riparian, and wetland communities. It appears to do this by outcompeting native species for light, nutrients, and water and perhaps by exuding allelopathic substances that inhibit the growth of other plants. In western North America, fennel can be found from the coastal and inland wildland-urban interface east into hill and mountain areas, excluding desert habitats. On Santa Cruz Island, California for example, fennel has achieved 50 to 90% absolute cover.

Production
As grouped by the United Nations Food and Agriculture Organization, production data for fennel are combined with similar spices – anise, star anise, and coriander. In 2014, India produced 60% of the world output of fennel, with China and Bulgaria as leading secondary producers (table).

Uses

Fennel was prized by the ancient Greeks and Romans, who used it as medicine, food, and insect repellent. Fennel tea was believed to give courage to the warriors before battle. According to Greek mythology, Prometheus used a giant stalk of fennel to carry fire from Mount Olympus to Earth. Emperor Charlemagne required the cultivation of fennel on all imperial farms.

Florence fennel is one of the three main herbs used in the preparation of absinthe, an alcoholic mixture which originated as a medicinal elixir in Europe and became, by the late 19th century, a popular alcoholic drink in France and other countries. Fennel fruit is a common and traditional spice in flavored Scandinavian brännvin (a loosely defined group of distilled spirits, which include akvavit). Fennel is also featured in the Chinese Materia Medica for its medicinal functions.

Nutrition
A raw fennel bulb is 90% water, 1% protein, 7% carbohydrates, and contains negligible fat.

Dried fennel seeds are typically used as a spice in minute quantities. A reference amount of 100 g of fennel seeds provides 345 kcal of food energy and is a rich source (20% or more of the Daily Value, DV) of protein, dietary fiber, B vitamins and several dietary minerals, especially calcium, iron, magnesium and manganese, all of which exceed 90% DV (table). Fennel seeds are 52% carbohydrates (including 40% dietary fiber), 15% fat, 16% protein, and 9% water (table).

Cuisine
The bulb, foliage, and fruits of the fennel plant are used in many of the culinary traditions of the world. The small flowers of wild fennel (known as fennel "pollen") are the most potent form of fennel, but also the most expensive. Dried fennel fruit is an aromatic, anise-flavored spice, brown or green when fresh, slowly turning a dull grey as the fruit ages. For cooking, green fruits are optimal. The leaves are delicately flavored and similar in shape to dill. The bulb is a crisp vegetable that can be sautéed, stewed, braised, grilled, or eaten raw. Tender young leaves are used for garnishes, as a salad, to add flavor to salads, to flavor sauces to be served with puddings, and in soups and fish sauce. Both the inflated leaf bases and the tender young shoots can be eaten like celery.

Fennel fruits are sometimes confused with those of anise, which are similar in taste and appearance, though smaller. Fennel is also a flavoring in some natural toothpastes. The fruits are used in cookery and sweet desserts.

Many cultures in India, Afghanistan, Iran, and the Middle East use fennel fruits in cooking. In Iraq, fennel seeds are used as an ingredient in nigella-flavored breads. It is one of the most important spices in Kashmiri cuisine and Gujarati cooking. In Indian cuisine, whole fennel seeds and fennel powder are used as a spice in various sweet and savory dishes. It is an essential ingredient in the Assamese/Bengali/Oriya spice mixture panch phoron and in Chinese five-spice powders. In many parts of India, roasted fennel fruits are consumed as mukhwas, an after-meal digestive and breath freshener (saunf), or candied as comfit. Fennel seeds are also often used as an ingredient in paan, a breath freshener most popularly consumed in India. In China, fennel stem and leaves are often ingredients in the stuffings of jiaozi, baozi, or pies, as well in cold dishes as a green vegetable. Fennel fruits are present in well-known mixed spices such as the five-spice powder or.

Fennel leaves are used in some parts of India as leafy green vegetables either by themselves or mixed with other vegetables, cooked to be served and consumed as part of a meal. In Syria and Lebanon, the young leaves are used to make a special kind of egg omelette (along with onions and flour) called ijjeh.

Many egg, fish, and other dishes employ fresh or dried fennel leaves. Florence fennel is a key ingredient in some Italian salads, or it can be braised and served as a warm side dish. It may be blanched or marinated, or cooked in risotto.

Fennel fruits are the primary flavor component in Italian sausage. In Spain, the stems of the fennel plant are used in the preparation of pickled eggplants, berenjenas de Almagro. A herbal tea or tisane can also be made from fennel.

On account of its aromatic properties, fennel fruit forms one of the ingredients of the well-known compound liquorice powder. In the Indian subcontinent, fennel fruits are eaten raw, sometimes with a sweetener.

Culture

The Greek name for fennel is marathon (μάραθον) or marathos (μάραθος), and the place of the famous battle of Marathon literally means a plain with fennel. The word is first attested in Mycenaean Linear B form as ma-ra-tu-wo. On tablets MY Ge 602, MY Ge 606 + fr., MY Ge 605 + 607 + frr. + 60Sa + 605b. In Hesiod's Theogony, Prometheus steals the ember of fire from the gods in a hollow fennel stalk.

As Old English finule, fennel is one of the nine plants invoked in the pagan Anglo-Saxon Nine Herbs Charm, recorded in the 10th century.

In the 15th century, Portuguese settlers on Madeira noticed the abundance of wild fennel and used the Portuguese word funcho (fennel) and the suffix -al to form the name of a new town, Funchal.

Henry Wadsworth Longfellow's 1842 poem "The Goblet of Life" repeatedly refers to the plant and mentions its purported ability to strengthen eyesight:

Above the lower plants, it towers, The Fennel with its yellow flowers; And in an earlier age than ours Was gifted with the wondrous powers Lost vision to restore.

title: Fennel


text:
Talk:Slovenian Armed Forces

Requested move 18 February 2015

 * The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section.

The result of the move request was: moved. Number   5  7  12:05, 26 February 2015 (UTC)

Military of Slovenia → Slovenian Armed Forces – Official name. See http://www.slovenskavojska.si/en/ Klemen Kocjancic (talk) 16:59, 18 February 2015 (UTC)
 * Support – It is both official, more correct, and more common. Move away. RGloucester  — ☎ 19:09, 20 February 2015 (UTC)

 * The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

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External links modified
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Slovenia does not have any Armed Forces
This article should be deleted because Slovenia does not have any armed forces.

title: Talk:Slovenian Armed Forces


text:
Talk:Morphological image processing

It'd probably be best to change the references to morphological operators so that the links refer to pages that won't require disambiguation. Perhaps something like 'Erosion (image processing)'.--Rethunk

In my opinion no way it should be merged with mathematical morphology! They refer to two totally different things. Zagyvasag 19:58, 23 June 2007 (UTC)

Since its foundation, the development of mathematical morphology has been driven mainly by image processing problems. Most publications on mathematical morphology are linked to image processing. Therefore keeping two articles will cause a lot of unnecessary redundancy. In my opinion, they should be merged.

title: Talk:Morphological image processing


text:
Shortstop

Shortstop, abbreviated SS, is the baseball or softball fielding position between second and third base, which is considered to be among the most demanding defensive positions. Historically, the position was assigned to defensive specialists who were typically poor at batting and were often placed at the bottom of the batting order. Today, shortstops are often able to hit well and many are placed at the top of the lineup. In the numbering system used by scorers to record defensive plays, the shortstop is assigned the number 6.

More hit balls go to the shortstop than to any other position, as there are more right-handed hitters in baseball than left-handed hitters, and most hitters have a tendency to pull the ball slightly. Like a second baseman, a shortstop must be agile, for example when performing a 4-6-3 double play. Also, like a third baseman, the shortstop fields balls hit to the left side of the infield, where a strong arm is needed to throw out a batter-runner before they reach the safety of first base.

History
Doc Adams of the Knickerbockers created the concept of the shortstop position, according to baseball historian John Thorn and Baseball Hall of Fame researcher Freddy Berowski. In the first five years the Knickerbockers played, the team fielded anywhere from eight to eleven players. The only infielders were the players covering each of the bases; if there were more than eight players, extra outfielders were sometimes used. The outfielders had difficulty throwing baseballs into the infield because of the balls' light weight. Adams's shortstop position, which he started playing at some time from 1849 to 1850, was used to field throws from the outfielders and throw to the three infielders. The position, more of a fourth outfielder than an infielder, was also then called "short fielder" (a term still used in soft-pitch softball for the 10th player).

With the advent of higher-quality baseballs, Adams moved to the infield, since the distance the balls could travel increased. However, Dickey Pearce, primarily of the Brooklyn Atlantics, is credited as the first to have played the shortstop position as it is played now. Adams had a long playing career with the Knickerbockers: he remained a player with the team until 1860.

Positioning
Unlike the pitcher and catcher, who must start every play in a designated area (the pitcher must be on the pitcher's mound, with one foot in contact with the pitcher's rubber, and the catcher must be behind home plate in the catcher's box) the shortstop and the other fielders can vary their positioning in response to what they anticipate will be the actions of the batter and runner(s) once the play begins.

The shortstop ordinarily is positioned near second base on the third-base side. Because right-handed hitters tend to hit the ball more toward third base, a shortstop will generally move closer to third base if the batter is batting right-handed, and more toward first base if the batter is batting left-handed. A shortstop typically has a strong throwing arm, because he has a relatively long throw to first base, and often has less time in which to make a throw, given that the ground balls he fields have often traveled relatively far. A shortstop must also be extremely agile, because balls hit to or near the shortstop position are usually hit harder than to other infield positions.

Shortstops are required to cover second base in double play situations when the ball is hit to the second baseman or first baseman. They also cover second when a runner is attempting a stolen base, but only when a left-handed hitter is batting because the infield will respond to a left-handed batter by shifting toward first base, resulting in the shortstop being the infielder who is closest to second base. Shortstops also must cover third at various times, including the rotation play; the latter occurs when there are runners on first and second and a sacrifice bunt is attempted toward third base, requiring the third baseman to move in away from third base in order to field it. Shortstops generally are given precedence on catching pop-ups in the infield as well, so they end up calling off other players many times, although on deep pop-ups they generally fall back when called off by an outfielder. They often become the cutoff man on balls to any part of the outfield that are being directed towards third base and all balls to left and center field that are destined for second base. Depending on the system the shortstop may cut balls from left field heading home; however, this is usually the job of the third baseman.

The emphasis on defense makes the position unusually difficult to fill. Historically, a strong shortstop did not have to be a good hitter. Some of the weakest hitters in Major League Baseball have played the position, including Mario Mendoza, for whom George Brett popularized the eponymous Mendoza Line to describe a batting average below .200. Since the 1960s, however, such mediocre hitting has become rarer as teams increasingly demand players with ability to both field and hit.

In practice, a marginal fielder as a shortstop who hits well can be moved to almost any other position, especially second base or third base, whether early in their careers (examples: George Brett and Mike Schmidt were both tried early in their careers as shortstops) or later due to diminished fielding range, slower reflexes, weaker throwing arms, increased risk of injury, or co-existence with another dominant shortstop, as with Ernie Banks, Cal Ripken Jr., Alex Rodríguez, Michael Young, or Miguel Tejada.

Shortstops inducted to the Baseball Hall of Fame

The year in which the player was inducted is given in brackets after his name.

 * Luis Aparicio (1984)
 * Luke Appling (1964)
 * Dave Bancroft (1971)
 * Ernie Banks (1977)
 * Lou Boudreau (1970)
 * Joe Cronin (1956)
 * George Davis (1998)
 * Travis Jackson (1982)
 * Hughie Jennings (1945)
 * Derek Jeter (2020)
 * Barry Larkin (2012)
 * John Henry "Pop" Lloyd (1977)
 * Rabbit Maranville (1954)
 * Paul Molitor (2004) (Played some games at SS in 1978, 1979, and 1982)
 * Pee Wee Reese (1984)
 * Cal Ripken Jr. (2007)
 * Phil Rizzuto (1994)
 * Joe Sewell (1977)
 * Ozzie Smith (2002)
 * Joe Tinker (1946)
 * Alan Trammell (2018)
 * Arky Vaughan (1985)
 * Honus Wagner (1936)
 * Bobby Wallace (1953)
 * John Montgomery Ward (1964)
 * Willie Wells (1997)
 * George Wright (1937)
 * Robin Yount (1999)

Multiple Gold Glove Award winners

 * Ozzie Smith: 13
 * Omar Vizquel: 11
 * Luis Aparicio: 9
 * Mark Belanger: 8
 * Dave Concepción: 5
 * Derek Jeter: 5
 * Tony Fernández: 4
 * Jimmy Rollins: 4
 * Alan Trammell: 4
 * Brandon Crawford: 3
 * J. J. Hardy: 3
 * Barry Larkin: 3
 * Roy McMillan: 3
 * Rey Ordóñez: 3
 * Andrelton Simmons: 3
 * Gene Alley: 2
 * Larry Bowa: 2
 * Orlando Cabrera: 2
 * Don Kessinger: 2
 * Francisco Lindor: 2
 * Édgar Rentería: 2
 * Cal Ripken Jr.: 2
 * Alex Rodriguez: 2
 * Troy Tulowitzki: 2
 * Zoilo Versalles: 2
 * Maury Wills: 2

All-time single season assist leaders among shortstops

 * 1) Ozzie Smith: 621 (San Diego Padres, 1980)
 * 2) Glenn Wright: 601 (Pittsburgh Pirates, 1924)
 * 3) Dave Bancroft: 598 (Philadelphia Phillies/New York Giants, 1920)
 * 4) Tommy Thevenow: 597 (St. Louis Cardinals, 1926)
 * 5) Iván DeJesús: 595 (Chicago Cubs, 1977)
 * 6) Cal Ripken Jr.: 583 (Baltimore Orioles, 1984)
 * 7) Whitey Wietelmann: 581 (Boston Braves, 1943)
 * 8) Dave Bancroft: 579 (New York Giants, 1922)
 * 9) Rabbit Maranville: 574 (Boston Braves, 1914)
 * 10) Don Kessinger: 573 (Chicago Cubs, 1968)

Source: (does not list teams)

All-time single season putout leaders among shortstops

 * 1) Donie Bush: 425 (Detroit Tigers, 1914)
 * 2) Hughie Jennings: 425 (Baltimore Orioles [National League], 1895)
 * 3) Joe Cassidy: 408 (Washington Senators, 1905)
 * 4) Rabbit Maranville: 407 (Boston Braves, 1914)
 * 5) Dave Bancroft: 405 (New York Giants, 1922)
 * 6) Eddie Miller: 405 (Boston Braves, 1940)
 * 7) Monte Cross: 404 (Philadelphia Phillies, 1898)
 * 8) Dave Bancroft: 396 (New York Giants, 1921)
 * 9) Mickey Doolan: 395 (Philadelphia Phillies, 1906)
 * 10) Buck Weaver: 392 (Chicago White Sox, 1913)

All-time single-season fielding percentage leaders among shortstops

 * 1) Mike Bordick: .9982 (Baltimore Orioles, 2002)
 * 2) Cal Ripken Jr.: .9956 (Baltimore Orioles, 1990)
 * 3) Omar Vizquel: .9954 (Cleveland Indians, 2000)
 * 4) Rey Sánchez: .9941 (Kansas City Royals, 2000)
 * 5) Rey Ordóñez: .9938 (New York Mets, 1999)
 * 6) Omar Vizquel: .9933 (San Francisco Giants, 2006)
 * 7) Omar Vizquel: .9931 (Cleveland Indians, 1998)
 * 8) J. J. Hardy: .9923 (Baltimore Orioles, 2012)
 * 9) Tony Fernández .9919 (Toronto Blue Jays, 1989)
 * 10) Rey Sánchez: .9915 (Kansas City Royals, 2001)

Number of seasons with 100+ double plays turned at shortstop (among Hall of Fame shortstops)

 * Cal Ripken Jr.: 12
 * Phil Rizzuto: 8
 * Lou Boudreau: 6
 * Ozzie Smith: 5
 * Luke Appling: 4
 * Luis Aparicio: 3
 * Ernie Banks: 3
 * Travis Jackson: 3
 * Pee Wee Reese: 2
 * Arky Vaughan: 2
 * Dave Bancroft: 1
 * Joe Cronin: 1
 * Derek Jeter: 1
 * Rabbit Maranville: 1
 * Joe Sewell: 1
 * Alan Trammell: 1
 * Robin Yount: 1

Source: baseballreference.com

title: Shortstop


text:
User:NuclearWinner

I'm a technical program manager in the San Francisco Bay Area of California, USA. I've studied materials science and engineering at MIT and Stanford, and have worked in high tech for 20+ years. I'm an avid reader of non-fiction, mysteries, poetry, scientific and political writing, and works in the French language. My personal crusade is to further the development of sustainable ecologies and economies. So far, I've done squat in this area but I have the best of intentions.

I enjoy supplying "Most wanted articles", as long as they do not relate to Grammy Awards. Receiving a Wiki barnstar has been one of the high points of my life. I have never been involved in a Wikipedia flame or editing war so please don't be the one to tempt me.

Looking over my contributions (both original articles and major additions), my interests appear to be:

American culture Cake, Catch Me If You Can, Corporate title, Management, Sammy Davis Jr., Venture capital, Wedding

World geography Bloemfontein, History of the Marshall Islands, Kwajalein, Orange River

Literature Gothic novel, How to read a poem, Louisa May Alcott, Scholar

France and the French language Avignon, France, Petrarch, Walloon

Mexico Guanajuato, Leon, States of Mexico

Health, medicine, and birth Calculus (dental), Fertility, Live birth, Vagina

Science and technology Precious metal

Politics, philosophy, public policy and law Just war, Adverse inference

Other topics 1912 Summer Olympics, Peruvian Paso

title: User:NuclearWinner


text:
User:Jptwo

interested in music, public vs. private domains, mathematics, civil rights, and more.

title: User:Jptwo


text:
Kris Kringle

Kris Kringle may refer to:

 * Santa Claus, by assimilation in the United States of the separate German tradition of the Christkindl
 * Kris Kringle, the lead character in Miracle on 34th Street
 * Kris Kringle, published male model Karun Randhawa
 * Kris Kringle, the lead character in Santa Claus Is Comin' to Town (film)
 * Kris Kringle, the father of the lead character in the 2019 movie Noelle
 * Chris Kringle, the lead character in the movie Fatman
 * Christkind (also called Christkindl), the Austrian and German Christmas gift-bringer, the Christ Child
 * Secret Santa, a gift exchange deriving from the Christkindl tradition
 * Danish chimney sweep/famous thief.

title: Kris Kringle


text:
English novel

The English novel is an important part of English literature. This article mainly concerns novels, written in English, by novelists who were born or have spent a significant part of their lives in England, Scotland, Wales, or Northern Ireland (or any part of Ireland before 1922). However, given the nature of the subject, this guideline has been applied with common sense, and reference is made to novels in other languages or novelists who are not primarily British, where appropriate.

Early novels in English
Historically, the English novel has generally been seen as beginning with Daniel Defoe's Robinson Crusoe (1719) and Moll Flanders (1722), though modern scholarship cites Aphra Behn's Love-Letters Between a Nobleman and His Sister (1684) John Bunyan's The Pilgrim's Progress (1678) and Aphra Behn's Oroonoko (1688) as more likely contenders, while earlier works such as Sir Thomas Malory's Morte d'Arthur (1485), and even the "Prologue" to Geoffrey Chaucer's Canterbury Tales (c. 1400) have been suggested. Another important early novel is Gulliver's Travels (1726, amended 1735), by Irish writer and clergyman Jonathan Swift, which is both a satire of human nature, as well as a parody of travellers' tales like Robinson Crusoe. The rise of the novel as an important literary genre is generally associated with the growth of the middle class in England.

Other major 18th-century English novelists are Samuel Richardson (1689–1761), author of the epistolary novels Pamela, or Virtue Rewarded (1740) and Clarissa (1747–48); Henry Fielding (1707–1754), who wrote Joseph Andrews (1742) and The History of Tom Jones, a Foundling (1749); Laurence Sterne (1713–1768), who published Tristram Shandy in parts between 1759 and 1767; Oliver Goldsmith (1728–1774), author of The Vicar of Wakefield (1766); Tobias Smollett (1721–1771), a Scottish novelist best known for his comic picaresque novels, such as The Adventures of Peregrine Pickle (1751) and The Expedition of Humphry Clinker (1771), who influenced Charles Dickens; and Fanny Burney (1752–1840), whose novels "were enjoyed and admired by Jane Austen," wrote Evelina (1778), Cecilia (1782) and Camilla (1796).

A noteworthy aspect of both the 18th- and 19th- century novel is the way the novelist directly addressed the reader. For example, the author might interrupt his or her narrative to pass judgment on a character, or pity or praise another, and inform or remind the reader of some other relevant issue.

Romantic period
The phrase 'Romantic novel' has several possible meanings. Here it refers to novels written during the Romantic era in literary history, which runs from the late 18th century until the beginning of the Victorian era in 1837. But to complicate matters there are novels written in the romance tradition by novelists like Walter Scott, Nathaniel Hawthorne, George Meredith. In addition the phrase today is mostly used to refer to the popular pulp-fiction genre that focusses on romantic love. The Romantic period is especially associated with the poets William Blake, William Wordsworth, Samuel Taylor Coleridge, George Byron, Percy Shelley and John Keats, though two major novelists, Jane Austen and Walter Scott, also published in the early 19th century.

Horace Walpole's 1764 novel, The Castle of Otranto, invented the Gothic fiction genre. The word gothic was originally used in the sense of medieval. This genre combines "the macabre, fantastic, and supernatural" and usually involves haunted castles, graveyards and various picturesque elements. Later novelist Ann Radcliffe introduced the brooding figure of the Gothic villain which developed into the Byronic hero. Her most popular and influential work, The Mysteries of Udolpho (1794), is frequently described as the archetypal Gothic novel. Vathek (1786), by William Beckford, and The Monk (1796), by Matthew Lewis, were further notable early works in both the Gothic and horror genres.

Mary Shelley's novel Frankenstein (1818), as another important Gothic novel as well as being an early example of science fiction. The vampire genre fiction began with John William Polidori's The Vampyre (1819). This short story was inspired by the life of Lord Byron and his poem The Giaour. An important later work is Varney the Vampire (1845), where many standard vampire conventions originated: Varney has fangs, leaves two puncture wounds on the neck of his victims, and has hypnotic powers and superhuman strength. Varney was also the first example of the "sympathetic vampire", who loathes his condition but is a slave to it.

Among more minor novelists in this period Maria Edgeworth (1768–1849) and Thomas Love Peacock (1785–1866) are worthy of comment. Edgeworth's novel Castle Rackrent (1800) is "the first fully developed regional novel in English" as well as "the first true historical novel in English" and an important influence on Walter Scott. Peacock was primarily a satirist in novels such as Nightmare Abbey (1818) and The Misfortunes of Elphin (1829).

Jane Austen's (1775–1817) works critique the novels of sensibility of the second half of the 18th century and are part of the transition to 19th-century realism. Her plots, though fundamentally comic, highlight the dependence of women on marriage to secure social standing and economic security. Austen brings to light the hardships women faced, who usually did not inherit money, could not work and where their only chance in life depended on the man they married. She reveals not only the difficulties women faced in her day, but also what was expected of men and of the careers they had to follow. This she does with wit and humour and with endings where all characters, good or bad, receive exactly what they deserve. Her work brought her little personal fame and only a few positive reviews during her lifetime, but the publication in 1869 of her nephew's A Memoir of Jane Austen introduced her to a wider public, and by the 1940s she had become accepted as a major writer. The second half of the 20th century saw a proliferation of Austen scholarship and the emergence of a Janeite fan culture. Austen's works include Pride and Prejudice (1813) Sense and Sensibility (1811), Mansfield Park, Persuasion and Emma.

The other major novelist at the beginning of the early 19th century was Sir Walter Scott (1771–1832), who was not only a highly successful British novelist but "the greatest single influence on fiction in the 19th century ... [and] a European figure". Scott established the genre of the historical novel with his series of Waverley Novels, including Waverley (1814), The Antiquary (1816), and The Heart of Midlothian (1818). However, Austen is today widely read and the source for films and television series, while Scott is less often read.

Victorian novel
It was in the Victorian era (1837–1901) that the novel became the leading literary genre in English. A number of women novelists were successful in the 19th century, although they often had to use a masculine pseudonym. At the beginning of the 19th century most novels were published in three volumes. However, monthly serialization was revived with the publication of Charles Dickens' Pickwick Papers in twenty parts between April 1836 and November 1837. Demand was high for each episode to introduce some new element, whether it was a plot twist or a new character, so as to maintain the readers' interest. Both Dickens and Thackeray frequently published this way.

In the 1830s and '40s, novelists began to show the influence of social critics on their work, especially Thomas Carlyle, who raised the "Condition-of-England Question" to describe "the social and political upheavals which followed the Reform Act of 1832". In response, novelists wrote "Condition of England novels", which were in many ways a reaction to rapid industrialization, and the social, political and economic issues associated with it, and were a means of commenting on abuses of government and industry and the suffering of the poor, who were not profiting from England's economic prosperity. Stories of the working-class poor were directed toward the middle class to help create sympathy and promote change. An early example is Charles Dickens' Oliver Twist (1837–38).

Charles Dickens emerged on the literary scene in the 1830s with the two novels already mentioned. Dickens wrote vividly about London life and struggles of the poor, but in a good-humoured fashion, accessible to readers of all classes. One of his most popular works to this day is A Christmas Carol (1843). In more recent years Dickens has been most admired for his later novels, such as Dombey and Son (1846–48), Great Expectations (1860–61), Bleak House (1852–53), Little Dorrit (1855–57), and Our Mutual Friend (1864–65). An early rival to Dickens was William Makepeace Thackeray, who during the Victorian period ranked second only to him, but he is now much less read and is known almost exclusively for Vanity Fair (1847). In that novel he satirizes whole swaths of humanity while retaining a light touch. It features his most memorable character, the engagingly roguish Becky Sharp.

The Brontë sisters were other significant novelists in the 1840s and 1850s. Their novels caused a sensation when they were first published and were subsequently accepted as classics. They had written compulsively from early childhood and were first published, at their own expense, in 1846 as poets under the pseudonyms Currer, Ellis, and Acton Bell. The sisters returned to prose, producing a novel each the following year: Charlotte's Jane Eyre, Emily's Wuthering Heights and Anne's Agnes Grey. Later, Anne's The Tenant of Wildfell Hall (1848) and Charlotte's Villette (1853) were published. Elizabeth Gaskell was also a successful writer and her first novel, Mary Barton, was published anonymously in 1848. Gaskell's North and South contrasts the lifestyle in the industrial north of England with the wealthier south. Even though her writing conforms to Victorian conventions, Gaskell usually frames her stories as critiques of contemporary attitudes: her early works focused on factory work in the Midlands. She always emphasised the role of women, with complex narratives and dynamic female characters.

Anthony Trollope (1815–82) was one of the most successful, prolific, and respected English novelists of the Victorian era. Some of his best-loved works are set in the imaginary county of Barsetshire, including The Warden (1855) and Barchester Towers (1857). He also wrote perceptive novels on political, social, and gender issues, and on other topical matters, including The Way with Live Now (1875). Trollope's novels portrayed the lives of the landowning and professional classes of early Victorian England.

George Eliot's (Mary Ann Evans) (1819–80) first novel Adam Bede was published in 1859. Her works, especially Middlemarch 1871–72), are important examples of literary realism, and are admired for their combination of high Victorian literary detail combined with an intellectual breadth that removes them from the narrow geographic confines they often depict.

An interest in rural matters and the changing social and economic situation of the countryside is seen in the novels of Thomas Hardy (1840–1928). A Victorian realist, in the tradition of George Eliot, he was also influenced both in his novels and poetry by Romanticism, especially by William Wordsworth. Charles Darwin is another important influence on Thomas Hardy. Like Charles Dickens he was also highly critical of much in Victorian society, though Hardy focused more on a declining rural society. While Hardy wrote poetry throughout his life and regarded himself primarily as a poet, his first collection was not published until 1898, so that initially he gained fame as the author of such novels as, Far from the Madding Crowd (1874), The Mayor of Casterbridge (1886), Tess of the d'Urbervilles (1891), and Jude the Obscure (1895). He ceased writing novels following adverse criticism of this last novel. In novels such as The Mayor of Casterbridge and Tess of the d'Urbervilles Hardy attempts to create modern works in the genre of tragedy, that are modelled on the Greek drama, especially Aeschylus and Sophocles, though in prose, not poetry, a novel not drama, and with characters of low social standing, not nobility. Another significant late 19th-century novelist is George Gissing (1857–1903) who published 23 novels between 1880 and 1903. His best known novel is New Grub Street (1891).

Important developments occurred in genre fiction in this era. Although pre-dated by John Ruskin's The King of the Golden River in 1841, the history of the modern fantasy genre is generally said to begin with George MacDonald, the influential author of The Princess and the Goblin and Phantastes (1858). William Morris was a popular English poet who also wrote several fantasy novels during the latter part of the nineteenth century. Wilkie Collins' epistolary novel The Moonstone (1868), is generally considered the first detective novel in the English language, while The Woman in White is regarded as one of the finest sensation novels. H. G. Wells's (1866–1946) writing career began in the 1890s with science fiction novels like The Time Machine (1895), and The War of the Worlds (1898) which describes an invasion of late Victorian England by Martians. Wells is seen, along with Frenchman Jules Verne (1828–1905), as inventing the scientific romance. He also wrote realistic fiction about the lower middle class in novels such as Kipps (1905) and The History of Mr Polly (1910).

20th century
The major novelists writing in Britain at the start of the 20th century were an Irishman James Joyce (1882–1941) and two immigrants, American Henry James (1843–1916) and Pole Joseph Conrad (1857–1924). The modernist tradition in the novel, with its emphasis "towards the ever more minute and analytic exposition of mental life", begins with James and Conrad, in novels such as The Ambassadors (1903), The Golden Bowl (1904) and Lord Jim (1900). Other important early modernists were Dorothy Richardson (1873–1957), whose novel Pointed Roof (1915), is one of the earliest example of the stream of consciousness technique, and D. H. Lawrence (1885–1930), who wrote with understanding about the social life of the lower and middle classes and the personal life of those who could not adapt to the social norms of his time. Sons and Lovers (1913), is widely regarded as his earliest masterpiece. There followed The Rainbow (1915), though it was immediately seized by the police, and its sequel Women in Love published in 1920. Lawrence attempted to explore human emotions more deeply than his contemporaries and challenged the boundaries of the acceptable treatment of sexual issues, most notably in Lady Chatterley's Lover, which was privately published in Florence in 1928. However, the unexpurgated version of this novel was not published until 1959. In 1922 Irishman James Joyce's important modernist novel Ulysses appeared. Set during one day in Dublin in June 1904, the novel has been called "a demonstration and summation of the entire movement." In it Joyce creates parallels with Homer's epic poem the Odyssey.

Another significant modernist in the 1920 was Virginia Woolf (1882–1941), who was an influential feminist, member of the Bloomsbury Group, and a major stylistic innovator associated with the stream-of-consciousness technique. Her novels include Mrs Dalloway (1925), To the Lighthouse (1927), Orlando: A Biography (1928), and The Waves (1931). Her essay collection A Room of One's Own (1929) contains her famous dictum: "A woman must have money and a room of her own if she is to write fiction".

But while modernism was to become an important literary movement in the early decades of the new century, there were also many fine novelists who were not strictly modernists. These include E. M. Forster ((1879–1970), John Galsworthy ((1867–1933) (Nobel Prize in Literature, 1932), whose novels include The Forsyte Saga, Arnold Bennett (1867–1931) author of The Old Wives' Tale, and H. G. Wells (1866–1946), though Forster's work is "frequently regarded as containing both modernist and Victorian elements". E. M. Forster's A Passage to India (1924), reflected challenges to imperialism, while his earlier works, such as A Room with a View (1908) and Howards End (1910), examined the restrictions and hypocrisy of Edwardian society in England. The most popular British writer of the early years of the 20th century was arguably Rudyard Kipling (1865–1936), a highly versatile writer of novels, short stories and poems and to date the youngest ever recipient of the Nobel Prize for Literature (1907).

A significant English writer in the 1930s and 1940s was George Orwell (1903–50), who is especially remembered for his satires of totalitarianism, Nineteen Eighty-Four (1949) and Animal Farm (1945). Evelyn Waugh (1903–66) satirised the "bright young things" of the 1920s and 1930s, notably in A Handful of Dust (1934) and Decline and Fall (1928), while Brideshead Revisited (1945) has a theological basis, setting out to examine the effect of divine grace on its main characters. Irishwoman and Bloomsbury Group member Elizabeth Bowen is known for her novels about the Irish Protestant gentry, such as The Death of the Heart (1938) and London during World War II bombing raids, The Heat of the Day (1948). Aldous Huxley (1894–1963) published his famous dystopia Brave New World in 1932, the same year as John Cowper Powys's (1872–1963) A Glastonbury Romance. Samuel Beckett (1906–89) published his first major work, the novel Murphy in 1938. This same year Graham Greene's (1904–91) first major novel Brighton Rock was published. Then in 1939 James Joyce published Finnegans Wake. In this work Joyce creates a special language to express the consciousness of a character who is dreaming.

Graham Greene was an important novelist whose works span the 1930s to the 1980s. Greene was a convert to Catholicism and his novels explore the ambivalent moral and political issues of the modern world. Notable for an ability to combine serious literary acclaim with broad popularity, his novels include, The Heart of the Matter (1948), A Burnt-Out Case (1961), and The Human Factor (1978). Evelyn Waugh's (1903–1966) career also continued after World War II, and in "1961 he completed his most considerable work, a trilogy about the war entitled Sword of Honour. In 1947 Malcolm Lowry published Under the Volcano. One of the most influential novels of the immediate post-war period was William Cooper's (1910–2002) naturalistic Scenes from Provincial Life (1950), which was a conscious rejection of the modernist tradition. Other novelists writing in the 1950s and later were: Anthony Powell (1905–2000) whose twelve-volume cycle of novels A Dance to the Music of Time (1951–75), is a comic examination of movements and manners, power and passivity in English political, cultural, and military life in the mid-20th century; comic novelist Kingsley Amis is best known for his academic satire Lucky Jim (1954); Nobel Prize laureate William Golding's allegorical novel Lord of the Flies (1954), explores how culture created by man fails, using as an example a group of British schoolboys marooned on a deserted island. Philosopher Iris Murdoch was a prolific writer of novels that deal with such things as sexual relationships, morality, and the power of the unconscious. Her works include Under the Net (1954), The Black Prince (1973) and The Green Knight (1993). Scottish writer Muriel Spark also began publishing in the 1950s. She pushed the boundaries of realism in her novels. Her first, The Comforters (1957), concerns a woman who becomes aware that she is a character in a novel; The Prime of Miss Jean Brodie (1961), jumps forward at the end to reveal the fates that befell its characters. Anthony Burgess is especially remembered for his dystopian novel A Clockwork Orange (1962), set in the not-too-distant future, which was made into a film by Stanley Kubrick in 1971. In the entirely different genre of Gothic fantasy Mervyn Peake (1911–1968) published his highly successful Gormenghast trilogy between 1946 and 1959.

Immigrant authors played a major role in post-war literature. Doris Lessing (1919) from Southern Rhodesia (now Zimbabwe), published her first novel The Grass is Singing in 1950, after immigrating to England. She initially wrote about her African experiences. Lessing soon became a dominant presence in the English literary scene, frequently publishing right through the century, and won the Nobel prize for literature in 2007. Salman Rushdie (born 1947) is another among a number of post Second World War writers from the former British colonies who permanently settled in Britain. Rushdie achieved fame with Midnight's Children 1981, which was awarded both the James Tait Black Memorial Prize and Booker Prize, and named Booker of Bookers in 1993. His most controversial novel The Satanic Verses (1989), was inspired in part by the life of Muhammad. V. S. Naipaul (1932–2018), born in Trinidad, wrote among other things A House for Mr. Biswas (1961) and A Bend in the River (1979). Naipaul won the Nobel Prize in Literature. Also from the West Indies George Lamming (1927–1922) is best remembered for In the Castle of the Skin (1953). Another important immigrant writer Kazuo Ishiguro (born 1954) was born in Japan, but his parents immigrated to Britain when he was six. His works include The Remains of the Day {1989) and Never Let Me Go (2005). Ishiguro won the Nobel Prize in Literature in 2017.

Scotland has in the late 20th-century produced several important novelists, including James Kelman (born 1946), who like Samuel Beckett can create humour out of the most grim situations. How Late it Was, How Late (1994), won the Booker Prize that year; A. L. Kennedy (born 1965) whose 2007 novel Day was named Book of the Year in the Costa Book Awards. In 2007 she won the Austrian State Prize for European Literature; Alasdair Gray (1934–2019) whose Lanark: A Life in Four Books (1981) is a dystopian fantasy set in his home town Glasgow. Another contemporary Scot is Irvine Welsh, whose novel Trainspotting (1993), gives a brutal depiction of the lives of working class Edinburgh drug users.

Angela Carter (1940–1992) was a novelist and journalist, known for her feminist, magical realism, and picaresque works. Writing from the 1960s until the 1980s, her novels include, The Infernal Desire Machines of Doctor Hoffman (1972) and Nights at the Circus (1984). Margaret Drabble (born 1939) is a novelist, biographer, and critic, who has published from the 1960s until this century. Her older sister, A. S. Byatt (born 1936) is best known for Possession published in 1990.

Among popular novelists Daphne Du Maurier wrote Rebecca, a mystery novel, in 1938 and W. Somerset Maugham’s (1874–1965) Of Human Bondage (1915), a strongly autobiographical novel, is generally agreed to be his masterpiece. In genre fiction Agatha Christie was an important writer of crime novels, short stories, and plays, best remembered for her 80 detective novels and her successful West End theatre plays. Christie's novels include Murder on the Orient Express (1934), Death on the Nile (1937), and And Then There Were None (1939). Another popular writer during the Golden Age of detective fiction was Dorothy L. Sayers, while Georgette Heyer created the historical romance genre.

Contemporary novelists
Martin Amis (1949 to 2023) was one of the most prominent of contemporary British novelists. His best-known novels are Money (1984) and London Fields (1989). Pat Barker (born 1943) has won many awards for her fiction.

Novelist and screenwriter Ian McEwan (born 1948) is another of contemporary Britain's most highly regarded writers. His works include The Cement Garden (1978) and Enduring Love (1997), which was made into a film. In 1998 McEwan won the Booker Prize with Amsterdam, while Atonement (2001) was made into an Oscar-winning film. McEwan was awarded the Jerusalem Prize in 2011. Jeanette Winterson’s (born 1959) novels explore themes of sexuality and religion, such as Oranges Are Not the Only Fruit (1985) and Sexing the Cherry (1989).

Zadie Smith's (born 1975) Whitbread Book Award winning novel White Teeth (2000), mixes pathos and humour, focusing on the later lives of two war time friends in London. Julian Barnes (born 1946) is another successful living novelist, who won the 2011 Man Booker Prize for his book The Sense of an Ending, while three of his earlier books had been shortlisted for the Booker Prize.

Monica Ali came to prominence in 2003 with the acclaimed novel Brick Lane. A chronicler of everyday life in multicultural Britain, Ali was hailed by critics and awarded several awards for the novel including the Man Booker Prize for Fiction.

One of the more ambitious novelists to emerge in contemporary English literature is David Mitchell whose far-reaching novel Cloud Atlas (2004) spans from the 19th century into the future.

In the early 21st century an outstanding concern with historical fiction has been noted. Hilary Mantel (1952-2022) had success with the critically acclaimed historical novel Wolf Hall (2009).

Survey
In 2003 the BBC carried out a UK survey entitled The Big Read in order to find the "nation's best-loved novel" of all time, with works by English novelists Tolkien, Austen, Pullman, Adams and Rowling making up the top five on the list.

Famous novelists (alphabetical order)
• Amis, Kingsley

• Amis, Martin

• Austen, Jane

• Barnes, Julian

• Becket, Samuel

• Bowen, Elizabeth

• Brontë, Anne

• Brontë, Charlotte

• Brontë, Emily

• Burney, Fanny, later Madame D'Arblay

• Butler, Samuel

• Carroll, Lewis

• Collins, Wilkie

• Conan Doyle, Arthur

• Conrad, Joseph

• Defoe, Daniel

• Dickens, Charles

• Disraeli, Benjamin

• Du Maurier, Daphne

• Eliot, George

• Fielding, Henry

• Ford, Ford Madox

• Forster, E. M.

• Forster, Margaret

• Gaskell, Elizabeth

• Gissing, George

• Goldsmith, Oliver

• Greene, Graham

• Hardy, Thomas

• Huxley, Aldous

• Ishiguro, Kazuo

• James, Henry

• Joyce, James

• Kipling, Rudyard

• Lawrence, D. H.

• Lessing, Doris

• Lewis, C. S.

• Lewis, Wyndham

• Lowry, Malcolm

• Meredith, George

• Murdoch, Iris

• Naipaul, V. S.

• Oliphant, Margaret, traditionally known as Mrs Oliphant

• Orwell, George

• Powell, Anthony

• Powys, John Cowper

• Powys, T. F.

• Pullman, Philip

• Reade, Charles

• Richardson, Dorothy

• Richardson, Samuel

• Rushdie, Salman

• Sackville-West, Vita

• Scott, Walter

• Shelley, Mary

• Smith, Charlotte Turner

• Smollett, Tobias

• Sterne, Laurence

• Stevenson, Robert Louis

• Swift, Jonathan

• Thackeray, William

• Tolkien, J. R. R.

• Trollope, Anthony

• Ward, Mary, traditionally known as Mrs Humphry Ward

• Waugh, Evelyn

• Wells, H. G.

• Wilde, Oscar

• Winterson, Jeanette

• Woolf, Virginia

• Wyndham, John

title: English novel


text:
Talk:Dirigible

Redundant
This article is redundant with the article for "airship". I propose to merge the two, since "airship" is the less obscure term leaving just the explanation about the origin of the term and a pointer to the "airship" article. [blimpguy] - Fri Jan 24 01:43:34 GMT 2003

I disagree, there should be similar articles for each term with the differences made apparent. BTW, for what most people think of when hearing the term airship, dirigible would be the proper term and in that case airship should point to dirigible but I feel there is enough of a difference to warrant two similar but separate articles. I will work on the dirigible article myself over time.

title: Talk:Dirigible