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Dorado

Dorado is a constellation in the Southern Sky. It was named in the late 16th century and is now one of the 88 modern constellations. Its name refers to the mahi-mahi (Coryphaena hippurus), which is known as dorado ("golden") in Spanish, although it has also been depicted as a swordfish. Dorado contains most of the Large Magellanic Cloud, the remainder being in the constellation Mensa. The South Ecliptic pole also lies within this constellation.

Even though the name Dorado is not Latin but Spanish, astronomers give it the Latin genitive form Doradus when naming its stars; it is treated (like the adjacent asterism Argo Navis) as a feminine proper name of Greek origin ending in -ō (like Io or Callisto or Argo), which have a genitive ending -ūs.

History
Dorado was one of twelve constellations named by Petrus Plancius from the observations of Pieter Dirkszoon Keyser and Frederick de Houtman. It appeared: Dorado represents a dolphinfish; it has also been called the goldfish because Dorado are gold-colored.
 * On a 35 cm celestial globe published in 1597 (or 1598) in Amsterdam by Plancius with Jodocus Hondius.
 * First depiction in a celestial atlas, in Johann Bayer's Uranometria of 1603.
 * In Johannes Kepler's edition of Tycho Brahe's star list in the Rudolphine Tables of 1627; this was the first time that it was given the alternative name Xiphias, the swordfish. The name Dorado ultimately became dominant and was adopted by the IAU.

Stars
Alpha Doradus is a blue-white star of magnitude 3.3, 176 light-years from Earth. It is the brightest star in Dorado. Beta Doradus is a notably bright Cepheid variable star. It is a yellow-tinged supergiant star that has a minimum magnitude of 4.1 and a maximum magnitude of 3.5. One thousand and forty light-years from Earth, Beta Doradus has a period of 9 days and 20 hours.

R Doradus is one of the many variable stars in Dorado. S Dor, 9.721 hypergiant in the Large Magellanic Cloud, is the prototype of S Doradus variable stars. The variable star R Doradus 5.73 has the largest-known apparent size of any star other than the Sun. Gamma Doradus is the prototype of the Gamma Doradus variable stars.

Supernova 1987A was the closest supernova to occur since the invention of the telescope. SNR 0509-67.5 is the remnant of an unusually energetic Type 1a supernova from about 400 years ago.

HE 0437-5439 is a hypervelocity star escaping from the Milky Way/Magellanic Cloud system.

Dorado is also the location of the South Ecliptic pole, which lies near the fish's head. The pole was called "Polus Doradinalis" by Philipp von Zesen, aka Caesius.

In early 2020, the exoplanet TOI-700 d was discovered orbiting the star TOI-700 in Dorado. This is the first potentially Earth-like exoplanet to be discovered by the Transiting Exoplanet Survey Satellite.

Deep-sky objects
Because Dorado contains part of the Large Magellanic Cloud, it is rich in deep sky objects. The Large Magellanic Cloud, 25,000 light-years in diameter, is a satellite galaxy of the Milky Way Galaxy, located at a distance of 179,000 light-years. It has been deformed by its gravitational interactions with the larger Milky Way. In 1987, it became host to SN 1987A, the first supernova of 1987 and the closest since 1604. This 25,000-light-year-wide galaxy contains over 10,000 million stars. All coordinates given are for Epoch J2000.0.

 * N 180B is an emission nebula located in the Large Magellanic Cloud.
 * NGC 1566 (RA 04h 20m 00s Dec -56° 56.3′) is a face-on spiral galaxy. It gives its name to the NGC 1566 Group of galaxies.
 * NGC 1755 (RA 04h 55m 13s Dec -68° 12.2′) is a globular cluster.
 * NGC 1763 (RA 04h 56m 49s Dec -68° 24.5′) is a bright nebula associated with three type B stars.
 * NGC 1761 (RA 04h 56m 37s Dec -66° 28.4') is a open cluster.
 * NGC 1820 (RA 05h 04m 02s Dec -67° 15.9′) is an open cluster.
 * NGC 1850 (RA 05h 08m 44s Dec -68° 45.7′) is a globular cluster.
 * NGC 1854 (RA 05h 09m 19s Dec -68° 50.8′) is a globular cluster.
 * NGC 1869 (RA 05h 13m 56s Dec -67° 22.8′) is an open cluster.
 * NGC 1901 (RA 05h 18m 15s Dec -68° 26.2′) is an open cluster.
 * NGC 1910 (RA 05h 18m 43s Dec -69° 13.9′) is an open cluster.
 * NGC 1936 (RA 05h 22m 14s Dec -67° 58.7′) is a bright nebula and is one of four NGC objects in close proximity, the others being NGC 1929, NGC 1934 and NGC 1935.
 * NGC 1978 (RA 05h 28m 36s Dec -66° 14.0′) is an open cluster.
 * NGC 2002 (RA 05h 30m 17s Dec -66° 53.1′) is an open cluster.
 * NGC 2014 (RA 05h 44m 12.7s Dec −67° 42′ 57″) is a red emission nebula.
 * NGC 2020 (RA 05h 44m 12.7s Dec −67° 42′ 57″) is an HII region surrounding a Wolf–Rayet star.
 * NGC 2027 (RA 05h 35m 00s Dec -66° 55.0′) is an open cluster.
 * NGC 2032 (RA 05h 35m 21s Dec -67° 34.1′; also known as "Seagull Nebula") is a nebula complex that contains four NGC designations: NGC 2029, NGC 2032, NGC 2035 and NGC 2040.
 * NGC 2074 (RA 05h 39m 03.0s Dec −69° 29′ 54″) is an emission nebula.
 * NGC 2078 (RA 05h 39m 54s Dec −69° 44′ 54″) is an emission nebula.
 * NGC 2080, also called the "Ghost Head Nebula", is an emission nebula that is 50 light-years wide in the Large Magellanic Cloud. It is named for the two distinct white patches that it possesses, which are regions of recent star formation. The western portion is colored green from doubly ionized oxygen, the southern portion is red from hydrogen alpha emissions, and the center region is colored yellow from both oxygen and hydrogen emissions. The western white patch, A1, has one massive, recently formed star inside. The eastern patch, A2, has several stars hidden in its dust.
 * Tarantula Nebula is in the Large Magellanic Cloud, named for its spiderlike shape. It is also designated 30 Doradus, as it is visible to the naked eye as a slightly out-of-focus star. Larger than any nebula in the Milky Way at 1,000 light-years in diameter, it is also brighter, because it is illuminated by the open star cluster NGC 2070, which has at its center the star cluster R136. The illuminating stars are supergiants.

 * NGC 2164 (RA 05h 58m 53s Dec -68° 30.9′) is a globular cluster.
 * N44 is a superbubble in the Large Magellanic Cloud that is 1,000 light-years wide. Its overall structure is shaped by the 40 hot stars towards its center. Within the superbubble of N44 is a smaller bubble catalogued as N44F. It is approximately 35 light-years in diameter and is shaped by an incredibly hot star at its center, which has a stellar wind speed of 7 million kilometers per hour. N44F also features dust columns with probable star formation hidden inside.

Equivalents
In Chinese astronomy, the stars of Dorado are in two of Xu Guangqi's Southern Asterisms (近南極星區, Jìnnánjíxīngōu): the White Patches Attached (夾白, Jiābái) and the Goldfish (金魚, Jīnyú).

Namesakes

 * Dorado (SS-248) and Dorado (SS-526), two United States Navy submarines, were named after the same sea creature as the constellation.

title: Dorado


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Draco (lawgiver)

Draco (Δράκων, Drakōn; fl. c. 625-600 BC), also called Drako or Drakon, was the first recorded legislator of Athens in Ancient Greece. He replaced the prevailing system of oral law and blood feud by the Draconian constitution, a written code to be enforced only by a court of law.

His name may be a metaphor (δράκων literally means "sharp-sighted"), not the name of an actual person, since there are reasons to believe that he is also a fiction, entirely or in part. This is supported by the lack of a patronymic and any biographical information.

Since the 19th century, the adjective draconian (Greek: δρακόντειος drakónteios) refers to similarly unforgiving rules or laws, in Greek, English, and other European languages.

Life
During the 39th Olympiad, in 622 or 621 BC, Draco established the legal code with which he is identified.

Little is known about Draco’s life. He may have belonged to the Greek nobility of Attica prior to the period of the Seven Sages of Greece, as per the 10th-century Byzantine encyclopedia, the Suda. The Suda chronicles a folkloric story of his death in the Aeginetan theatre: in a traditional ancient Greek show of approval, his supporters "threw so many hats and shirts and cloaks on his head that he suffocated, and was buried in that same theatre". The truth about his death is still unclear, but it is known that Draco was driven out of Athens by the Athenians to the neighbouring island of Aegina, where he spent the remainder of his life.

Draconian constitution
The laws (θεσμοί – thesmoi) that he laid were the first written constitution of Athens. So that no one would be unaware of them, they were posted on wooden tablets (ἄξονες – axones), where they were preserved for almost two centuries on steles of the shape of four-sided pyramids (κύρβεις – kyrbeis). The tablets were called axones, perhaps because they could be pivoted along the pyramid's axis to read any side.

The constitution featured several major innovations:
 * Instead of oral laws known to a special class, arbitrarily applied and interpreted, all laws were written, thus being made known to all literate citizens (who could appeal to the Areopagus for injustices): "the constitution formed under Draco, when the first code of laws was drawn up". (Aristotle: Athenian Constitution, Part 5, Section 41)
 * The laws distinguish between murder and involuntary homicide, a novel concept at that time.

The laws were particularly harsh. For example, any debtor whose status was lower than that of his creditor was forced into slavery. The punishment was more lenient for those owing a debt to a member of a lower class. The death penalty was the punishment for even minor offences, such as stealing a cabbage. Concerning the liberal use of the death penalty in the Draconic code, Plutarch states:

"It is said that Drakon himself, when asked why he had fixed the punishment of death for most offences, answered that he considered these lesser crimes to deserve it, and he had no greater punishment for more important ones."

All Draco's laws were repealed by Solon in the early 6th century BC, with the exception of the homicide law.

Homicide law
After much debate, the Athenians decided to revise the laws, including the homicide law, in 409 BC. The homicide law is a highly fragmented inscription, but states that it is up to the victim's relatives to prosecute a killer.

According to the preserved part of the inscription, unintentional homicides received a sentence of exile. It is not clear whether Draco's law specified the punishment for intentional homicide. In 409 BC, intentional homicide was punished by death, but Draco's law begins: "καὶ ἐὰμ μὲ ‘κ [π]ρονοί[α]ς [κ]τ[ένει τίς τινα, φεύγ]ε[ν]." Although ambiguous and difficult to translate, one suggested translation is: "Even if a man not intentionally kills another, he is exiled."

Council of Four Hundred
Draco introduced the lot-chosen Council of Four Hundred, distinct from the Areopagus, which evolved in later constitutions to play a large role in Athenian democracy. Aristotle notes that Draco, while having the laws written, merely legislated for an existing unwritten Athenian constitution such as setting exact qualifications for eligibility for office.

According to Aristotle, Draco extended the franchise to all free men who could furnish themselves with a set of military equipment. However, this claim is not based on the authentic tradition, thus untrue as claimed by Welwei in 1998. They elected the Council of Four Hundred from among their number; nine archons and the treasurers were drawn from persons possessing an unencumbered property of not less than ten minas, the generals (strategoi) and commanders of cavalry (hipparchoi) from those who could show an unencumbered property of not less than a hundred minas and had children born in lawful wedlock over ten years of age. Thus, in the event of their death, their estate could pass to a competent heir. These officers were required to hold to account the prytanes (councillors), strategoi (generals) and hipparchoi (cavalry officers) of the preceding year until their accounts had been audited. "The Council of Areopagus was guardian of the laws, and kept watch over the magistrates to see that they executed their offices in accordance with the laws. Any person who felt himself wronged might lay an information before the Council of Areopagus, on declaring what law was broken by the wrong done to him. But, as has been said before, loans were secured upon the persons of the debtors, and the land was in the hands of a few."

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Draco (lawgiver)


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Determinant

In mathematics, the determinant is a scalar-valued function of the entries of a square matrix. The determinant of a matrix $A$ is commonly denoted $det(A)$, $det A$, or $|A|$. Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the corresponding linear map is an isomorphism.

The determinant is completely determined by the two following properties: the determinant of a product of matrices is the product of their determinants, and the determinant of a triangular matrix is the product of its diagonal entries.

The determinant of a $2 × 2$ matrix is
 * $$\begin{vmatrix} a & b\\c & d \end{vmatrix}=ad-bc,$$

and the determinant of a $3 × 3$ matrix is
 * $$ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix}= aei + bfg + cdh - ceg - bdi - afh.$$

The determinant of an $n × n$ matrix can be defined in several equivalent ways, the most common being Leibniz formula, which expresses the determinant as a sum of $$n!$$ (the factorial of $n$) signed products of matrix entries. It can be computed by the Laplace expansion, which expresses the determinant as a linear combination of determinants of submatrices, or with Gaussian elimination, which allows computing a row echelon form with the same determinant, equal to the product of the diagonal entries of the row echelon form.

Determinants can also be defined by some of their properties. Namely, the determinant is the unique function defined on the $n × n$ matrices that has the four following properties:
 * 1) The determinant of the identity matrix is $1$.
 * 2) The exchange of two rows multiplies the determinant by $−1$.
 * 3) Multiplying a row by a number multiplies the determinant by this number.
 * 4) Adding a multiple of one row to another row does not change the determinant.

The above properties relating to rows (properties 2–4) may be replaced by the corresponding statements with respect to columns.

The determinant is invariant under matrix similarity. This implies that, given a linear endomorphism of a finite-dimensional vector space, the determinant of the matrix that represents it on a basis does not depend on the chosen basis. This allows defining the determinant of a linear endomorphism, which does not depends on the choice of a coordinate system.

Determinants occur throughout mathematics. For example, a matrix is often used to represent the coefficients in a system of linear equations, and determinants can be used to solve these equations (Cramer's rule), although other methods of solution are computationally much more efficient. Determinants are used for defining the characteristic polynomial of a square matrix, whose roots are the eigenvalues. In geometry, the signed $n$-dimensional volume of a $n$-dimensional parallelepiped is expressed by a determinant, and the determinant of a linear endomorphism determines how the orientation and the $n$-dimensional volume are transformed under the endomorphism. This is used in calculus with exterior differential forms and the Jacobian determinant, in particular for changes of variables in multiple integrals.

Two by two matrices
The determinant of a $2 × 2$ matrix $$\begin{pmatrix} a & b \\c & d \end{pmatrix}$$ is denoted either by "$det$" or by vertical bars around the matrix, and is defined as
 * $$\det \begin{pmatrix} a & b \\c & d \end{pmatrix} = \begin{vmatrix} a & b \\c & d \end{vmatrix} = ad - bc.$$

For example,
 * $$\det \begin{pmatrix} 3 & 7 \\1 & -4 \end{pmatrix} = \begin{vmatrix} 3 & 7 \\ 1 & {-4} \end{vmatrix} = (3 \cdot (-4)) - (7 \cdot 1) = -19.$$

First properties
The determinant has several key properties that can be proved by direct evaluation of the definition for $$2 \times 2$$-matrices, and that continue to hold for determinants of larger matrices. They are as follows: first, the determinant of the identity matrix $$\begin{pmatrix}1 & 0 \\ 0 & 1 \end{pmatrix}$$ is 1. Second, the determinant is zero if two rows are the same:
 * $$\begin{vmatrix} a & b \\ a & b \end{vmatrix} = ab - ba = 0.$$

This holds similarly if the two columns are the same. Moreover,
 * $$\begin{vmatrix}a & b + b' \\ c & d + d' \end{vmatrix} = a(d+d')-(b+b')c = \begin{vmatrix}a & b\\ c & d \end{vmatrix} + \begin{vmatrix}a & b' \\ c & d' \end{vmatrix}.$$

Finally, if any column is multiplied by some number $$r$$ (i.e., all entries in that column are multiplied by that number), the determinant is also multiplied by that number:
 * $$\begin{vmatrix} r \cdot a & b \\ r \cdot c & d \end{vmatrix} = rad - brc = r(ad-bc) = r \cdot \begin{vmatrix} a & b \\c & d \end{vmatrix}.$$

Geometric meaning

If the matrix entries are real numbers, the matrix $A$ can be used to represent two linear maps: one that maps the standard basis vectors to the rows of $A$, and one that maps them to the columns of $A$. In either case, the images of the basis vectors form a parallelogram that represents the image of the unit square under the mapping. The parallelogram defined by the rows of the above matrix is the one with vertices at $(0, 0)$, $(a, b)$, $(a + c, b + d)$, and $(c, d)$, as shown in the accompanying diagram.

The absolute value of $ad − bc$ is the area of the parallelogram, and thus represents the scale factor by which areas are transformed by $A$. (The parallelogram formed by the columns of $A$ is in general a different parallelogram, but since the determinant is symmetric with respect to rows and columns, the area will be the same.)

The absolute value of the determinant together with the sign becomes the oriented area of the parallelogram. The oriented area is the same as the usual area, except that it is negative when the angle from the first to the second vector defining the parallelogram turns in a clockwise direction (which is opposite to the direction one would get for the identity matrix).

To show that $ad − bc$ is the signed area, one may consider a matrix containing two vectors $u ≡ (a, b)$ and $v ≡ (c, d)$ representing the parallelogram's sides. The signed area can be expressed as $|u| |v| sin θ$ for the angle θ between the vectors, which is simply base times height, the length of one vector times the perpendicular component of the other. Due to the sine this already is the signed area, yet it may be expressed more conveniently using the cosine of the complementary angle to a perpendicular vector, e.g. $u^{⊥} = (−b, a)$, so that $|u^{⊥}| |v| cos θ′$ becomes the signed area in question, which can be determined by the pattern of the scalar product to be equal to $ad − bc$ according to the following equations:

 * $$\text{Signed area} =

|\boldsymbol{u}|\,|\boldsymbol{v}|\,\sin\,\theta = \left|\boldsymbol{u}^\perp\right|\,\left|\boldsymbol{v}\right|\,\cos\,\theta' = \begin{pmatrix} -b \\ a \end{pmatrix} \cdot \begin{pmatrix} c \\ d \end{pmatrix} = ad - bc. $$

Thus the determinant gives the scaling factor and the orientation induced by the mapping represented by A. When the determinant is equal to one, the linear mapping defined by the matrix is equi-areal and orientation-preserving.

The object known as the bivector is related to these ideas. In 2D, it can be interpreted as an oriented plane segment formed by imagining two vectors each with origin $(0, 0)$, and coordinates $(a, b)$ and $(c, d)$. The bivector magnitude (denoted by $(a, b) ∧ (c, d)$) is the signed area, which is also the determinant $ad − bc$.

If an $n × n$ real matrix A is written in terms of its column vectors $$A = \left[\begin{array}{c|c|c|c} \mathbf{a}_1 & \mathbf{a}_2 & \cdots & \mathbf{a}_n\end{array}\right]$$, then

A\begin{pmatrix}1 \\ 0\\ \vdots \\0\end{pmatrix} = \mathbf{a}_1, \quad A\begin{pmatrix}0 \\ 1\\ \vdots \\0\end{pmatrix} = \mathbf{a}_2, \quad \ldots, \quad A\begin{pmatrix}0 \\0 \\ \vdots \\1\end{pmatrix} = \mathbf{a}_n. $$

This means that $$A$$ maps the unit n-cube to the n-dimensional parallelotope defined by the vectors $$\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_n,$$ the region $$P = \left\{c_1 \mathbf{a}_1 + \cdots + c_n\mathbf{a}_n \mid 0 \leq c_i\leq 1 \ \forall i\right\}.$$

The determinant gives the signed n-dimensional volume of this parallelotope, $$\det(A) = \pm \text{vol}(P),$$ and hence describes more generally the n-dimensional volume scaling factor of the linear transformation produced by A. (The sign shows whether the transformation preserves or reverses orientation.) In particular, if the determinant is zero, then this parallelotope has volume zero and is not fully n-dimensional, which indicates that the dimension of the image of A is less than n. This means that A produces a linear transformation which is neither onto nor one-to-one, and so is not invertible.

Definition
Let A be a square matrix with n rows and n columns, so that it can be written as
 * $$A = \begin{bmatrix}

a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots &  \vdots \\ a_{n,1} & a_{n,2} & \cdots & a_{n,n} \end{bmatrix}.$$

The entries $$a_{1,1}$$ etc. are, for many purposes, real or complex numbers. As discussed below, the determinant is also defined for matrices whose entries are in a commutative ring.

The determinant of A is denoted by det(A), or it can be denoted directly in terms of the matrix entries by writing enclosing bars instead of brackets:
 * $$\begin{vmatrix}

a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots &  \vdots \\ a_{n,1} & a_{n,2} & \cdots & a_{n,n} \end{vmatrix}.$$

There are various equivalent ways to define the determinant of a square matrix A, i.e. one with the same number of rows and columns: the determinant can be defined via the Leibniz formula, an explicit formula involving sums of products of certain entries of the matrix. The determinant can also be characterized as the unique function depending on the entries of the matrix satisfying certain properties. This approach can also be used to compute determinants by simplifying the matrices in question.

3 × 3 matrices
The Leibniz formula for the determinant of a $3 × 3$ matrix is the following:
 * $$\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}

= aei + bfg + cdh - ceg - bdi - afh.\ $$ In this expression, each term has one factor from each row, all in different columns, arranged in increasing row order. For example, bdi has b from the first row second column, d from the second row first column, and i from the third row third column. The signs are determined by how many transpositions of factors are necessary to arrange the factors in increasing order of their columns (given that the terms are arranged left-to-right in increasing row order): positive for an even number of transpositions and negative for an odd number. For the example of bdi, the single transposition of bd to db gives dbi, whose three factors are from the first, second and third columns respectively; this is an odd number of transpositions, so the term appears with negative sign. The rule of Sarrus is a mnemonic for the expanded form of this determinant: the sum of the products of three diagonal north-west to south-east lines of matrix elements, minus the sum of the products of three diagonal south-west to north-east lines of elements, when the copies of the first two columns of the matrix are written beside it as in the illustration. This scheme for calculating the determinant of a $3 × 3$ matrix does not carry over into higher dimensions.

n × n matrices
Generalizing the above to higher dimensions, the determinant of an $$n \times n$$ matrix is an expression involving permutations and their signatures. A permutation of the set $$\{1, 2, \dots, n \}$$ is a bijective function $$\sigma$$ from this set to itself, with values $$\sigma(1), \sigma(2),\ldots,\sigma(n)$$ exhausting the entire set. The set of all such permutations, called the symmetric group, is commonly denoted $$S_n$$. The signature $$\sgn(\sigma)$$ of a permutation $$\sigma$$ is $$+1,$$ if the permutation can be obtained with an even number of transpositions (exchanges of two entries); otherwise, it is $$-1.$$

Given a matrix
 * $$A=\begin{bmatrix}

a_{1,1}\ldots a_{1,n}\\ \vdots\qquad\vdots\\ a_{n,1}\ldots a_{n,n} \end{bmatrix},$$ the Leibniz formula for its determinant is, using sigma notation for the sum,
 * $$\det(A)=\begin{vmatrix}

a_{1,1}\ldots a_{1,n}\\ \vdots\qquad\vdots\\ a_{n,1}\ldots a_{n,n} \end{vmatrix} = \sum_{\sigma \in S_n}\sgn(\sigma)a_{1,\sigma(1)}\cdots a_{n,\sigma(n)}.$$

Using pi notation for the product, this can be shortened into
 * $$\det(A) = \sum_{\sigma \in S_n} \left( \sgn(\sigma) \prod_{i=1}^n a_{i,\sigma(i)}\right)$$.

The Levi-Civita symbol $$\varepsilon_{i_1,\ldots,i_n}$$ is defined on the $n$-tuples of integers in $$\{1,\ldots,n\}$$ as $0$ if two of the integers are equal, and otherwise as the signature of the permutation defined by the n-tuple of integers. With the Levi-Civita symbol, the Leibniz formula becomes
 * $$\det(A) = \sum_{i_1,i_2,\ldots,i_n} \varepsilon_{i_1\cdots i_n} a_{1,i_1} \!\cdots a_{n,i_n},$$

where the sum is taken over all $n$-tuples of integers in $$\{1,\ldots,n\}.$$

Characterization of the determinant
The determinant can be characterized by the following three key properties. To state these, it is convenient to regard an $$n \times n$$-matrix A as being composed of its $$n$$ columns, so denoted as
 * $$A = \big ( a_1, \dots, a_n \big ),$$

where the column vector $$a_i$$ (for each i) is composed of the entries of the matrix in the i-th column.

&= \big | a_1, \dots, a_{j-1}, r \cdot v + w, a_{j+1}, \dots, a_n | \\ &= r \cdot | a_1, \dots, v, \dots a_n | + | a_1, \dots, w, \dots, a_n | \end{align}$$
 * 1)  $$\det\left(I\right) = 1$$, where $$I$$ is an identity matrix.
 * 2)  The determinant is multilinear: if the jth column of a matrix $$A$$ is written as a linear combination $$a_j = r \cdot v + w$$ of two column vectors v and w and a number r, then the determinant of A is expressible as a similar linear combination:
 * $$\begin{align}|A|
 * 1) The determinant is alternating: whenever two columns of a matrix are identical, its determinant is 0:
 * $$| a_1, \dots, v, \dots, v, \dots, a_n| = 0.$$

If the determinant is defined using the Leibniz formula as above, these three properties can be proved by direct inspection of that formula. Some authors also approach the determinant directly using these three properties: it can be shown that there is exactly one function that assigns to any $$n \times n$$-matrix A a number that satisfies these three properties. This also shows that this more abstract approach to the determinant yields the same definition as the one using the Leibniz formula.

To see this it suffices to expand the determinant by multi-linearity in the columns into a (huge) linear combination of determinants of matrices in which each column is a standard basis vector. These determinants are either 0 (by property 9) or else ±1 (by properties 1 and 12 below), so the linear combination gives the expression above in terms of the Levi-Civita symbol. While less technical in appearance, this characterization cannot entirely replace the Leibniz formula in defining the determinant, since without it the existence of an appropriate function is not clear.

Immediate consequences
These rules have several further consequences:
 * The determinant is a homogeneous function, i.e., $$\det(cA) = c^n\det(A)$$ (for an $$n \times n$$ matrix $$A$$).
 * Interchanging any pair of columns of a matrix multiplies its determinant by −1. This follows from the determinant being multilinear and alternating (properties 2 and 3 above): $$|a_1, \dots, a_j, \dots a_i, \dots, a_n| = - |a_1, \dots, a_i, \dots, a_j, \dots, a_n|.$$ This formula can be applied iteratively when several columns are swapped. For example $$|a_3, a_1, a_2, a_4 \dots, a_n| = - |a_1, a_3, a_2, a_4, \dots, a_n| = |a_1, a_2, a_3, a_4, \dots, a_n|.$$ Yet more generally, any permutation of the columns multiplies the determinant by the sign of the permutation.
 * If some column can be expressed as a linear combination of the other columns (i.e. the columns of the matrix form a linearly dependent set), the determinant is 0. As a special case, this includes: if some column is such that all its entries are zero, then the determinant of that matrix is 0.
 * Adding a scalar multiple of one column to another column does not change the value of the determinant. This is a consequence of multilinearity and being alternative: by multilinearity the determinant changes by a multiple of the determinant of a matrix with two equal columns, which determinant is 0, since the determinant is alternating.
 * If $$A$$ is a triangular matrix, i.e. $$a_{ij}=0$$, whenever $$i>j$$ or, alternatively, whenever $$i<j$$, then its determinant equals the product of the diagonal entries: $$\det(A) = a_{11} a_{22} \cdots a_{nn} = \prod_{i=1}^n a_{ii}.$$ Indeed, such a matrix can be reduced, by appropriately adding multiples of the columns with fewer nonzero entries to those with more entries, to a diagonal matrix (without changing the determinant). For such a matrix, using the linearity in each column reduces to the identity matrix, in which case the stated formula holds by the very first characterizing property of determinants. Alternatively, this formula can also be deduced from the Leibniz formula, since the only permutation $$\sigma$$ which gives a non-zero contribution is the identity permutation.

Example
These characterizing properties and their consequences listed above are both theoretically significant, but can also be used to compute determinants for concrete matrices. In fact, Gaussian elimination can be applied to bring any matrix into upper triangular form, and the steps in this algorithm affect the determinant in a controlled way. The following concrete example illustrates the computation of the determinant of the matrix $$A$$ using that method:

 * $$A = \begin{bmatrix}

-2 & -1 & 2 \\ 2 & 1 & 4 \\ -3 & 3 & -1 \end{bmatrix}. $$

Combining these equalities gives $$|A| = -|E| = -(18 \cdot 3 \cdot (-1)) = 54.$$

Transpose
The determinant of the transpose of $$A$$ equals the determinant of A:
 * $$\det\left(A^\textsf{T}\right) = \det(A)$$.

This can be proven by inspecting the Leibniz formula. This implies that in all the properties mentioned above, the word "column" can be replaced by "row" throughout. For example, viewing an $n × n$ matrix as being composed of n rows, the determinant is an n-linear function.

Multiplicativity and matrix groups
The determinant is a multiplicative map, i.e., for square matrices $$A$$ and $$B$$ of equal size, the determinant of a matrix product equals the product of their determinants:
 * $$\det(AB) = \det (A) \det (B)$$

This key fact can be proven by observing that, for a fixed matrix $$B$$, both sides of the equation are alternating and multilinear as a function depending on the columns of $$A$$. Moreover, they both take the value $$\det B$$ when $$A$$ is the identity matrix. The above-mentioned unique characterization of alternating multilinear maps therefore shows this claim.

A matrix $$A$$ with entries in a field is invertible precisely if its determinant is nonzero. This follows from the multiplicativity of the determinant and the formula for the inverse involving the adjugate matrix mentioned below. In this event, the determinant of the inverse matrix is given by
 * $$\det\left(A^{-1}\right) = \frac{1}{\det(A)} = [\det(A)]^{-1}$$.

In particular, products and inverses of matrices with non-zero determinant (respectively, determinant one) still have this property. Thus, the set of such matrices (of fixed size $$n$$ over a field $$K$$) forms a group known as the general linear group $$\operatorname{GL}_n(K)$$ (respectively, a subgroup called the special linear group $$\operatorname{SL}_n(K) \subset \operatorname{GL}_n(K)$$. More generally, the word "special" indicates the subgroup of another matrix group of matrices of determinant one. Examples include the special orthogonal group (which if n is 2 or 3 consists of all rotation matrices), and the special unitary group.

Because the determinant respects multiplication and inverses, it is in fact a group homomorphism from $$\operatorname{GL}_n(K)$$ into the multiplicative group $$K^\times$$ of nonzero elements of $$K$$. This homomorphism is surjective and its kernel is $$\operatorname{SL}_n(K)$$ (the matrices with determinant one). Hence, by the first isomorphism theorem, this shows that $$\operatorname{SL}_n(K)$$ is a normal subgroup of $$\operatorname{GL}_n(K)$$, and that the quotient group $$\operatorname{GL}_n(K)/\operatorname{SL}_n(K)$$ is isomorphic to $$K^\times$$.

The Cauchy–Binet formula is a generalization of that product formula for rectangular matrices. This formula can also be recast as a multiplicative formula for compound matrices whose entries are the determinants of all quadratic submatrices of a given matrix.

Laplace expansion
Laplace expansion expresses the determinant of a matrix $$A$$ recursively in terms of determinants of smaller matrices, known as its minors. The minor $$M_{i,j}$$ is defined to be the determinant of the $$(n-1) \times (n-1)$$-matrix that results from $$A$$ by removing the $$i$$-th row and the $$j$$-th column. The expression $$(-1)^{i+j}M_{i,j}$$ is known as a cofactor. For every $$i$$, one has the equality
 * $$\det(A) = \sum_{j=1}^n (-1)^{i+j} a_{i,j} M_{i,j},$$

which is called the Laplace expansion along the $i$th row. For example, the Laplace expansion along the first row ($$i=1$$) gives the following formula:

\begin{vmatrix}a&b&c\\ d&e&f\\ g&h&i\end{vmatrix} = a\begin{vmatrix}e&f\\ h&i\end{vmatrix} - b\begin{vmatrix}d&f\\ g&i\end{vmatrix} + c\begin{vmatrix}d&e\\ g&h\end{vmatrix} $$ Unwinding the determinants of these $$2 \times 2$$-matrices gives back the Leibniz formula mentioned above. Similarly, the Laplace expansion along the $$j$$-th column is the equality
 * $$\det(A)= \sum_{i=1}^n (-1)^{i+j} a_{i,j} M_{i,j}.$$

Laplace expansion can be used iteratively for computing determinants, but this approach is inefficient for large matrices. However, it is useful for computing the determinants of highly symmetric matrix such as the Vandermonde matrix $$\begin{vmatrix} 1 &        1 &         1 & \cdots &         1 \\ x_1 &      x_2 &       x_3 & \cdots &       x_n \\ x_1^2 &    x_2^2 &     x_3^2 & \cdots &     x_n^2 \\ \vdots &   \vdots &    \vdots & \ddots &    \vdots \\ x_1^{n-1} & x_2^{n-1} & x_3^{n-1} & \cdots & x_n^{n-1} \end{vmatrix} = \prod_{1 \leq i < j \leq n} \left(x_j - x_i\right). $$The n-term Laplace expansion along a row or column can be generalized to write an n x n determinant as a sum of $$\tbinom nk$$ terms, each the product of the determinant of a k x k submatrix and the determinant of the complementary (n−k) x (n−k) submatrix.

Adjugate matrix
The adjugate matrix $$\operatorname{adj}(A)$$ is the transpose of the matrix of the cofactors, that is,
 * $$(\operatorname{adj}(A))_{i,j} = (-1)^{i+j} M_{ji}.$$

For every matrix, one has
 * $$(\det A) I = A\operatorname{adj}A = (\operatorname{adj}A)\,A. $$

Thus the adjugate matrix can be used for expressing the inverse of a nonsingular matrix:
 * $$A^{-1} = \frac 1{\det A}\operatorname{adj}A. $$

Block matrices
The formula for the determinant of a $$2 \times 2$$-matrix above continues to hold, under appropriate further assumptions, for a block matrix, i.e., a matrix composed of four submatrices $$A, B, C, D$$ of dimension $$m \times m$$, $$m \times n$$, $$n \times m$$ and $$n \times n$$, respectively. The easiest such formula, which can be proven using either the Leibniz formula or a factorization involving the Schur complement, is
 * $$\det\begin{pmatrix}A& 0\\ C& D\end{pmatrix} = \det(A) \det(D) = \det\begin{pmatrix}A& B\\ 0& D\end{pmatrix}.$$

If $$A$$ is invertible, then it follows with results from the section on multiplicativity that
 * $$\begin{align}

\det\begin{pmatrix}A& B\\ C& D\end{pmatrix} & = \det(A)\det\begin{pmatrix}A& B\\ C& D\end{pmatrix} \underbrace{\det\begin{pmatrix}A^{-1}& -A^{-1} B\\ 0& I_n\end{pmatrix}}_{=\,\det(A^{-1})\,=\,(\det A)^{-1}}\\ & = \det(A) \det\begin{pmatrix}I_m& 0\\ C A^{-1}& D-C A^{-1} B\end{pmatrix}\\ & = \det(A) \det(D - C A^{-1} B), \end{align}$$ which simplifies to $$\det (A) (D - C A^{-1} B)$$ when $$D$$ is a $$1 \times 1$$-matrix.

A similar result holds when $$D$$ is invertible, namely
 * $$\begin{align}

\det\begin{pmatrix}A& B\\ C& D\end{pmatrix} & = \det(D)\det\begin{pmatrix}A& B\\ C& D\end{pmatrix} \underbrace{\det\begin{pmatrix}I_m& 0\\ -D^{-1} C& D^{-1}\end{pmatrix}}_{=\,\det(D^{-1})\,=\,(\det D)^{-1}}\\ & = \det(D) \det\begin{pmatrix}A - B D^{-1} C& B D^{-1}\\ 0& I_n\end{pmatrix}\\ & = \det(D) \det(A - B D^{-1} C). \end{align}$$ Both results can be combined to derive Sylvester's determinant theorem, which is also stated below.

If the blocks are square matrices of the same size further formulas hold. For example, if $$C$$ and $$D$$ commute (i.e., $$CD=DC$$), then

 * $$\det\begin{pmatrix}A& B\\ C& D\end{pmatrix} = \det(AD - BC).$$

This formula has been generalized to matrices composed of more than $$2 \times 2$$ blocks, again under appropriate commutativity conditions among the individual blocks.

For $$A = D $$ and $$B = C$$, the following formula holds (even if $$A$$ and $$B$$ do not commute)
 * $$\det\begin{pmatrix}A& B\\ B& A\end{pmatrix} = \det(A - B) \det(A + B).$$

Sylvester's determinant theorem
Sylvester's determinant theorem states that for A, an $m × n$ matrix, and B, an $n × m$ matrix (so that A and B have dimensions allowing them to be multiplied in either order forming a square matrix):

 * $$\det\left(I_\mathit{m} + AB\right) = \det\left(I_\mathit{n} + BA\right),$$

where Im and In are the $m × m$ and $n × n$ identity matrices, respectively.

From this general result several consequences follow. 1. For the case of column vector c and row vector r, each with m components, the formula allows quick calculation of the determinant of a matrix that differs from the identity matrix by a matrix of rank 1:
 * $\det\left(I_\mathit{m} + cr\right) = 1 + rc.$

2. More generally, for any invertible $m × m$ matrix X,
 * $\det(X + AB) = \det(X) \det\left(I_\mathit{n} + BX^{-1}A\right),$

3. For a column and row vector as above:
 * $\det(X + cr) = \det(X) \det\left(1 + rX^{-1}c\right) = \det(X) + r\,\operatorname{adj}(X)\,c.$

4. For square matrices $A$ and $B$ of the same size, the matrices $AB$ and $BA$ have the same characteristic polynomials (hence the same eigenvalues).

Sum
The determinant of the sum $$A+B$$ of two square matrices of the same size is not in general expressible in terms of the determinants of A and of B.

However, for positive semidefinite matrices $$A$$, $$B$$ and $$C$$ of equal size, $$\det(A + B + C) + \det(C) \geq \det(A + C) + \det(B + C)\text{,}$$ with the corollary $$\det(A + B) \geq \det(A) + \det(B)\text{.}$$

Brunn–Minkowski theorem implies that the $n$th root of determinant is a concave function, when restricted to Hermitian positive-definite $$n\times n$$ matrices. Therefore, if $A$ and $B$ are Hermitian positive-definite $$n\times n$$ matrices, one has $$\sqrt[n]{\det(A+B)}\geq\sqrt[n]{\det(A)}+\sqrt[n]{\det(B)},$$ since the $n$th root of the determinant is a homogeneous function.

Sum identity for 2×2 matrices
For the special case of $$2\times 2$$ matrices with complex entries, the determinant of the sum can be written in terms of determinants and traces in the following identity:
 * $$\det(A+B) = \det(A) + \det(B) + \text{tr}(A)\text{tr}(B) - \text{tr}(AB).$$

$$

This has an application to $$2\times 2$$ matrix algebras. For example, consider the complex numbers as a matrix algebra. The complex numbers have a representation as matrices of the form $$aI + b\mathbf{i} := a\begin{pmatrix} 1 &       0 \\         0 & 1 \end{pmatrix} + b\begin{pmatrix} 0 &        -1 \\         1 & 0 \end{pmatrix}$$ with $$a$$ and $$b$$ real. Since $$\text{tr}(\mathbf{i}) = 0$$, taking $$A = aI$$ and $$B = b\mathbf{i}$$ in the above identity gives
 * $$\det(aI + b\mathbf{i}) = a^2\det(I) + b^2\det(\mathbf{i}) = a^2 + b^2.$$

This result followed just from $$\text{tr}(\mathbf{i}) = 0$$ and $$\det(I) = \det(\mathbf{i}) = 1$$.

Eigenvalues and characteristic polynomial
The determinant is closely related to two other central concepts in linear algebra, the eigenvalues and the characteristic polynomial of a matrix. Let $$A$$ be an $$n \times n$$-matrix with complex entries. Then, by the Fundamental Theorem of Algebra, $$A$$ must have exactly n eigenvalues $$\lambda_1, \lambda_2, \ldots, \lambda_n$$. (Here it is understood that an eigenvalue with algebraic multiplicity $μ$ occurs $μ$ times in this list.) Then, it turns out the determinant of $A$ is equal to the product of these eigenvalues,
 * $$\det(A) = \prod_{i=1}^n \lambda_i=\lambda_1\lambda_2\cdots\lambda_n.$$

The product of all non-zero eigenvalues is referred to as pseudo-determinant.

From this, one immediately sees that the determinant of a matrix $$A$$ is zero if and only if $$0$$ is an eigenvalue of $$A$$. In other words, $$A$$ is invertible if and only if $$0$$ is not an eigenvalue of $$A$$.

The characteristic polynomial is defined as
 * $$\chi_A(t) = \det(t \cdot I - A).$$

Here, $$t$$ is the indeterminate of the polynomial and $$I$$ is the identity matrix of the same size as $$A$$. By means of this polynomial, determinants can be used to find the eigenvalues of the matrix $$A$$: they are precisely the roots of this polynomial, i.e., those complex numbers $$\lambda$$ such that
 * $$\chi_A(\lambda) = 0.$$

A Hermitian matrix is positive definite if all its eigenvalues are positive. Sylvester's criterion asserts that this is equivalent to the determinants of the submatrices
 * $$A_k := \begin{bmatrix}

a_{1,1} & a_{1,2} & \cdots & a_{1,k} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,k} \\ \vdots & \vdots & \ddots &  \vdots \\ a_{k,1} & a_{k,2} & \cdots & a_{k,k} \end{bmatrix}$$

being positive, for all $$k$$ between $$1$$ and $$n$$.

Trace
The trace tr(A) is by definition the sum of the diagonal entries of $A$ and also equals the sum of the eigenvalues. Thus, for complex matrices $A$,
 * $$\det(\exp(A)) = \exp(\operatorname{tr}(A))$$

or, for real matrices $A$,
 * $$\operatorname{tr}(A) = \log(\det(\exp(A))).$$

Here exp($A$) denotes the matrix exponential of $A$, because every eigenvalue $λ$ of $A$ corresponds to the eigenvalue exp($λ$) of exp($A$). In particular, given any logarithm of $A$, that is, any matrix $L$ satisfying
 * $$\exp(L) = A$$

the determinant of $A$ is given by
 * $$\det(A) = \exp(\operatorname{tr}(L)).$$

For example, for $n = 2$, $n = 3$, and $n = 4$, respectively,
 * $$\begin{align}

\det(A) &= \frac{1}{2}\left(\left(\operatorname{tr}(A)\right)^2 - \operatorname{tr}\left(A^2\right)\right), \\ \det(A) &= \frac{1}{6}\left(\left(\operatorname{tr}(A)\right)^3 - 3\operatorname{tr}(A) ~ \operatorname{tr}\left(A^2\right) + 2 \operatorname{tr}\left(A^3\right)\right), \\ \det(A) &= \frac{1}{24}\left(\left(\operatorname{tr}(A)\right)^4 - 6\operatorname{tr}\left(A^2\right)\left(\operatorname{tr}(A)\right)^2 + 3\left(\operatorname{tr}\left(A^2\right)\right)^2 + 8\operatorname{tr}\left(A^3\right)~\operatorname{tr}(A) - 6\operatorname{tr}\left(A^4\right)\right). \end{align}$$

cf. Cayley-Hamilton theorem. Such expressions are deducible from combinatorial arguments, Newton's identities, or the Faddeev–LeVerrier algorithm. That is, for generic $n$, $detA = (−1)^{n}c_{0}$ the signed constant term of the characteristic polynomial, determined recursively from
 * $$c_n = 1; c_{n-m} = -\frac{1}{m}\sum_{k=1}^m c_{n-m+k} \operatorname{tr}\left(A^k\right) (1 \le m \le n)~.$$

In the general case, this may also be obtained from
 * $$\det(A) = \sum_{\begin{array}{c}k_1,k_2,\ldots,k_n \geq 0\\k_1+2k_2+\cdots+nk_n=n\end{array}}\prod_{l=1}^n \frac{(-1)^{k_l+1}}{l^{k_l}k_l!} \operatorname{tr}\left(A^l\right)^{k_l},$$

where the sum is taken over the set of all integers $k_{l} ≥ 0$ satisfying the equation
 * $$\sum_{l=1}^n lk_l = n.$$

The formula can be expressed in terms of the complete exponential Bell polynomial of n arguments sl = −(l – 1)! tr(Al) as
 * $$\det(A) = \frac{(-1)^n}{n!} B_n(s_1, s_2, \ldots, s_n).$$

This formula can also be used to find the determinant of a matrix $A^{I}_{J}$ with multidimensional indices $I = (i_{1}, i_{2}, ..., i_{r})$ and $J = (j_{1}, j_{2}, ..., j_{r})$. The product and trace of such matrices are defined in a natural way as
 * $$(AB)^I_J = \sum_K A^I_K B^K_J, \operatorname{tr}(A) = \sum_I A^I_I.$$

An important arbitrary dimension $n$ identity can be obtained from the Mercator series expansion of the logarithm when the expansion converges. If every eigenvalue of A is less than 1 in absolute value,
 * $$\det(I + A) = \sum_{k=0}^\infty \frac{1}{k!} \left(-\sum_{j=1}^\infty \frac{(-1)^j}{j} \operatorname{tr}\left(A^j\right)\right)^k\,,$$

where $I$ is the identity matrix. More generally, if
 * $$\sum_{k=0}^\infty \frac{1}{k!} \left(-\sum_{j=1}^\infty \frac{(-1)^j s^j}{j}\operatorname{tr}\left(A^j\right)\right)^k\,,$$

is expanded as a formal power series in $s$ then all coefficients of $s$$m$ for $m > n$ are zero and the remaining polynomial is $det(I + sA)$.

Upper and lower bounds
For a positive definite matrix $A$, the trace operator gives the following tight lower and upper bounds on the log determinant
 * $$\operatorname{tr}\left(I - A^{-1}\right) \le \log\det(A) \le \operatorname{tr}(A - I)$$

with equality if and only if $A = I$. This relationship can be derived via the formula for the Kullback-Leibler divergence between two multivariate normal distributions.

Also,
 * $$\frac{n}{\operatorname{tr}\left(A^{-1}\right)} \leq \det(A)^\frac{1}{n} \leq \frac{1}{n}\operatorname{tr}(A) \leq \sqrt{\frac{1}{n}\operatorname{tr}\left(A^2\right)}.$$

These inequalities can be proved by expressing the traces and the determinant in terms of the eigenvalues. As such, they represent the well-known fact that the harmonic mean is less than the geometric mean, which is less than the arithmetic mean, which is, in turn, less than the root mean square.

Derivative
The Leibniz formula shows that the determinant of real (or analogously for complex) square matrices is a polynomial function from $$\mathbf R^{n \times n}$$ to $$\mathbf R$$. In particular, it is everywhere differentiable. Its derivative can be expressed using Jacobi's formula:

 * $$\frac{d \det(A)}{d \alpha} = \operatorname{tr}\left(\operatorname{adj}(A) \frac{d A}{d \alpha}\right).$$

where $$\operatorname{adj}(A)$$ denotes the adjugate of $$A$$. In particular, if $$A$$ is invertible, we have

 * $$\frac{d \det(A)}{d \alpha} = \det(A) \operatorname{tr}\left(A^{-1} \frac{d A}{d \alpha}\right).$$

Expressed in terms of the entries of $$A$$, these are

 * $$ \frac{\partial \det(A)}{\partial A_{ij}}= \operatorname{adj}(A)_{ji} = \det(A)\left(A^{-1}\right)_{ji}.$$

Yet another equivalent formulation is

 * $$\det(A + \epsilon X) - \det(A) = \operatorname{tr}(\operatorname{adj}(A) X) \epsilon + O\left(\epsilon^2\right) = \det(A) \operatorname{tr}\left(A^{-1} X\right) \epsilon + O\left(\epsilon^2\right)$$,

using big O notation. The special case where $$A = I$$, the identity matrix, yields

 * $$\det(I + \epsilon X) = 1 + \operatorname{tr}(X) \epsilon + O\left(\epsilon^2\right).$$

This identity is used in describing Lie algebras associated to certain matrix Lie groups. For example, the special linear group $$\operatorname{SL}_n$$ is defined by the equation $$\det A = 1$$. The above formula shows that its Lie algebra is the special linear Lie algebra $$\mathfrak{sl}_n$$ consisting of those matrices having trace zero.

Writing a $$3 \times 3$$-matrix as $$A = \begin{bmatrix}a & b & c\end{bmatrix}$$ where $$a, b,c$$ are column vectors of length 3, then the gradient over one of the three vectors may be written as the cross product of the other two:

 * $$\begin{align}

\nabla_\mathbf{a}\det(A) &= \mathbf{b} \times \mathbf{c} \\ \nabla_\mathbf{b}\det(A) &= \mathbf{c} \times \mathbf{a} \\ \nabla_\mathbf{c}\det(A) &= \mathbf{a} \times \mathbf{b}. \end{align}$$

History
Historically, determinants were used long before matrices: A determinant was originally defined as a property of a system of linear equations. The determinant "determines" whether the system has a unique solution (which occurs precisely if the determinant is non-zero). In this sense, determinants were first used in the Chinese mathematics textbook The Nine Chapters on the Mathematical Art (九章算術, Chinese scholars, around the 3rd century BCE). In Europe, solutions of linear systems of two equations were expressed by Cardano in 1545 by a determinant-like entity.

Determinants proper originated separately from the work of Seki Takakazu in 1683 in Japan and parallelly of Leibniz in 1693. stated, without proof, Cramer's rule. Both Cramer and also were led to determinants by the question of plane curves passing through a given set of points.

Vandermonde (1771) first recognized determinants as independent functions. gave the general method of expanding a determinant in terms of its complementary minors: Vandermonde had already given a special case. Immediately following, Lagrange (1773) treated determinants of the second and third order and applied it to questions of elimination theory; he proved many special cases of general identities.

Gauss (1801) made the next advance. Like Lagrange, he made much use of determinants in the theory of numbers. He introduced the word "determinant" (Laplace had used "resultant"), though not in the present signification, but rather as applied to the discriminant of a quantic. Gauss also arrived at the notion of reciprocal (inverse) determinants, and came very near the multiplication theorem.

The next contributor of importance is Binet (1811, 1812), who formally stated the theorem relating to the product of two matrices of m columns and n rows, which for the special case of $m = n$ reduces to the multiplication theorem. On the same day (November 30, 1812) that Binet presented his paper to the Academy, Cauchy also presented one on the subject. (See Cauchy–Binet formula.) In this he used the word "determinant" in its present sense, summarized and simplified what was then known on the subject, improved the notation, and gave the multiplication theorem with a proof more satisfactory than Binet's. With him begins the theory in its generality.

used the functional determinant which Sylvester later called the Jacobian. In his memoirs in Crelle's Journal for 1841 he specially treats this subject, as well as the class of alternating functions which Sylvester has called alternants. About the time of Jacobi's last memoirs, Sylvester (1839) and Cayley began their work. introduced the modern notation for the determinant using vertical bars.

The study of special forms of determinants has been the natural result of the completion of the general theory. Axisymmetric determinants have been studied by Lebesgue, Hesse, and Sylvester; persymmetric determinants by Sylvester and Hankel; circulants by Catalan, Spottiswoode, Glaisher, and Scott; skew determinants and Pfaffians, in connection with the theory of orthogonal transformation, by Cayley; continuants by Sylvester; Wronskians (so called by Muir) by Christoffel and Frobenius; compound determinants by Sylvester, Reiss, and Picquet; Jacobians and Hessians by Sylvester; and symmetric gauche determinants by Trudi. Of the textbooks on the subject Spottiswoode's was the first. In America, Hanus (1886), Weld (1893), and Muir/Metzler (1933) published treatises.

Cramer's rule
Determinants can be used to describe the solutions of a linear system of equations, written in matrix form as $$Ax = b$$. This equation has a unique solution $$x$$ if and only if $$\det (A)$$ is nonzero. In this case, the solution is given by Cramer's rule:
 * $$x_i = \frac{\det(A_i)}{\det(A)} \qquad i = 1, 2, 3, \ldots, n$$

where $$A_i$$ is the matrix formed by replacing the $$i$$-th column of $$A$$ by the column vector $$b$$. This follows immediately by column expansion of the determinant, i.e.
 * $$\det(A_i) =

\det\begin{bmatrix}a_1 & \ldots & b & \ldots & a_n\end{bmatrix} = \sum_{j=1}^n x_j\det\begin{bmatrix}a_1 & \ldots & a_{i-1} & a_j & a_{i+1} & \ldots & a_n\end{bmatrix} = x_i\det(A) $$ where the vectors $$a_j$$ are the columns of A. The rule is also implied by the identity

 * $$A\, \operatorname{adj}(A) = \operatorname{adj}(A)\, A = \det(A)\, I_n.$$

Cramer's rule can be implemented in $$\operatorname O(n^3)$$ time, which is comparable to more common methods of solving systems of linear equations, such as LU, QR, or singular value decomposition.

Linear independence
Determinants can be used to characterize linearly dependent vectors: $$\det A$$ is zero if and only if the column vectors (or, equivalently, the row vectors) of the matrix $$A$$ are linearly dependent. For example, given two linearly independent vectors $$v_1, v_2 \in \mathbf R^3$$, a third vector $$v_3$$ lies in the plane spanned by the former two vectors exactly if the determinant of the $$3 \times 3$$-matrix consisting of the three vectors is zero. The same idea is also used in the theory of differential equations: given functions $$f_1(x), \dots, f_n(x)$$ (supposed to be $$n-1$$ times differentiable), the Wronskian is defined to be
 * $$W(f_1, \ldots, f_n)(x) =

\begin{vmatrix} f_1(x) &        f_2(x) & \cdots &         f_n(x) \\ f_1'(x) &       f_2'(x) & \cdots &        f_n'(x) \\ \vdots &        \vdots & \ddots &         \vdots \\ f_1^{(n-1)}(x) & f_2^{(n-1)}(x) & \cdots & f_n^{(n-1)}(x) \end{vmatrix}.$$

It is non-zero (for some $$x$$) in a specified interval if and only if the given functions and all their derivatives up to order $$n-1$$ are linearly independent. If it can be shown that the Wronskian is zero everywhere on an interval then, in the case of analytic functions, this implies the given functions are linearly dependent. See the Wronskian and linear independence. Another such use of the determinant is the resultant, which gives a criterion when two polynomials have a common root.

Orientation of a basis
The determinant can be thought of as assigning a number to every sequence of n vectors in Rn, by using the square matrix whose columns are the given vectors. The determinant will be nonzero if and only if the sequence of vectors is a basis for Rn. In that case, the sign of the determinant determines whether the orientation of the basis is consistent with or opposite to the orientation of the standard basis. In the case of an orthogonal basis, the magnitude of the determinant is equal to the product of the lengths of the basis vectors. For instance, an orthogonal matrix with entries in Rn represents an orthonormal basis in Euclidean space, and hence has determinant of ±1 (since all the vectors have length 1). The determinant is +1 if and only if the basis has the same orientation. It is −1 if and only if the basis has the opposite orientation.

More generally, if the determinant of A is positive, A represents an orientation-preserving linear transformation (if A is an orthogonal $2 × 2$ or $3 × 3$ matrix, this is a rotation), while if it is negative, A switches the orientation of the basis.

Volume and Jacobian determinant
As pointed out above, the absolute value of the determinant of real vectors is equal to the volume of the parallelepiped spanned by those vectors. As a consequence, if $$f : \mathbf R^n \to \mathbf R^n$$ is the linear map given by multiplication with a matrix $$A$$, and $$S \subset \mathbf R^n$$ is any measurable subset, then the volume of $$f(S)$$ is given by $$|\det(A)|$$ times the volume of $$S$$. More generally, if the linear map $$f : \mathbf R^n \to \mathbf R^m$$ is represented by the $$m \times n$$-matrix $$A$$, then the $$n$$-dimensional volume of $$f(S)$$ is given by:
 * $$\operatorname{volume}(f(S)) = \sqrt{\det\left(A^\textsf{T} A\right)} \operatorname{volume}(S).$$

By calculating the volume of the tetrahedron bounded by four points, they can be used to identify skew lines. The volume of any tetrahedron, given its vertices $$a, b, c, d$$, $$\frac 1 6 \cdot |\det(a-b,b-c,c-d)|$$, or any other combination of pairs of vertices that form a spanning tree over the vertices.

For a general differentiable function, much of the above carries over by considering the Jacobian matrix of f. For
 * $$f: \mathbf R^n \rightarrow \mathbf R^n,$$

the Jacobian matrix is the $n × n$ matrix whose entries are given by the partial derivatives
 * $$D(f) = \left(\frac {\partial f_i}{\partial x_j}\right)_{1 \leq i, j \leq n}.$$

Its determinant, the Jacobian determinant, appears in the higher-dimensional version of integration by substitution: for suitable functions f and an open subset U of Rn (the domain of f), the integral over f(U) of some other function $φ : R^{n} → R^{m}$ is given by
 * $$\int_{f(U)} \phi(\mathbf{v})\, d\mathbf{v} = \int_U \phi(f(\mathbf{u})) \left|\det(\operatorname{D}f)(\mathbf{u})\right| \,d\mathbf{u}.$$

The Jacobian also occurs in the inverse function theorem.

When applied to the field of Cartography, the determinant can be used to measure the rate of expansion of a map near the poles.

Determinant of an endomorphism
The above identities concerning the determinant of products and inverses of matrices imply that similar matrices have the same determinant: two matrices A and B are similar, if there exists an invertible matrix X such that $A = X^{−1}BX$. Indeed, repeatedly applying the above identities yields

 * $$\det(A) = \det(X)^{-1} \det(B)\det(X) = \det(B) \det(X)^{-1} \det(X) = \det(B).$$

The determinant is therefore also called a similarity invariant. The determinant of a linear transformation
 * $$T : V \to V$$

for some finite-dimensional vector space V is defined to be the determinant of the matrix describing it, with respect to an arbitrary choice of basis in V. By the similarity invariance, this determinant is independent of the choice of the basis for V and therefore only depends on the endomorphism T.

Square matrices over commutative rings
The above definition of the determinant using the Leibniz rule holds works more generally when the entries of the matrix are elements of a commutative ring $$R$$, such as the integers $$\mathbf Z$$, as opposed to the field of real or complex numbers. Moreover, the characterization of the determinant as the unique alternating multilinear map that satisfies $$\det(I) = 1$$ still holds, as do all the properties that result from that characterization.

A matrix $$A \in \operatorname{Mat}_{n \times n}(R)$$ is invertible (in the sense that there is an inverse matrix whose entries are in $$R$$) if and only if its determinant is an invertible element in $$R$$. For $$R = \mathbf Z$$, this means that the determinant is +1 or −1. Such a matrix is called unimodular.

The determinant being multiplicative, it defines a group homomorphism
 * $$\operatorname{GL}_n(R) \rightarrow R^\times, $$

between the general linear group (the group of invertible $$n \times n$$-matrices with entries in $$R$$) and the multiplicative group of units in $$R$$. Since it respects the multiplication in both groups, this map is a group homomorphism.

Given a ring homomorphism $$f : R \to S$$, there is a map $$\operatorname{GL}_n(f) : \operatorname{GL}_n(R) \to \operatorname{GL}_n(S)$$ given by replacing all entries in $$R$$ by their images under $$f$$. The determinant respects these maps, i.e., the identity
 * $$f(\det((a_{i,j}))) = \det ((f(a_{i,j})))$$

holds. In other words, the displayed commutative diagram commutes.

For example, the determinant of the complex conjugate of a complex matrix (which is also the determinant of its conjugate transpose) is the complex conjugate of its determinant, and for integer matrices: the reduction modulo $$m$$ of the determinant of such a matrix is equal to the determinant of the matrix reduced modulo $$m$$ (the latter determinant being computed using modular arithmetic). In the language of category theory, the determinant is a natural transformation between the two functors $$\operatorname{GL}_n$$ and $$(-)^\times$$. Adding yet another layer of abstraction, this is captured by saying that the determinant is a morphism of algebraic groups, from the general linear group to the multiplicative group,
 * $$\det: \operatorname{GL}_n \to \mathbb G_m.$$

Exterior algebra
The determinant of a linear transformation $$T : V \to V$$ of an $$n$$-dimensional vector space $$V$$ or, more generally a free module of (finite) rank $$n$$ over a commutative ring $$R$$ can be formulated in a coordinate-free manner by considering the $$n$$-th exterior power $$\bigwedge^n V$$ of $$V$$. The map $$T$$ induces a linear map
 * $$\begin{align}

\bigwedge^n T: \bigwedge^n V &\rightarrow \bigwedge^n V \\ v_1 \wedge v_2 \wedge \dots \wedge v_n &\mapsto T v_1 \wedge T v_2 \wedge \dots \wedge T v_n. \end{align}$$

As $$\bigwedge^n V$$ is one-dimensional, the map $$\bigwedge^n T$$ is given by multiplying with some scalar, i.e., an element in $$R$$. Some authors such as use this fact to define the determinant to be the element in $$R$$ satisfying the following identity (for all $$v_i \in V$$):
 * $$\left(\bigwedge^n T\right)\left(v_1 \wedge \dots \wedge v_n\right) = \det(T) \cdot v_1 \wedge \dots \wedge v_n.$$

This definition agrees with the more concrete coordinate-dependent definition. This can be shown using the uniqueness of a multilinear alternating form on $$n$$-tuples of vectors in $$R^n$$. For this reason, the highest non-zero exterior power $$\bigwedge^n V$$ (as opposed to the determinant associated to an endomorphism) is sometimes also called the determinant of $$V$$ and similarly for more involved objects such as vector bundles or chain complexes of vector spaces. Minors of a matrix can also be cast in this setting, by considering lower alternating forms $$\bigwedge^k V$$ with $$k < n$$.

Generalizations and related notions
Determinants as treated above admit several variants: the permanent of a matrix is defined as the determinant, except that the factors $$\sgn(\sigma)$$ occurring in Leibniz's rule are omitted. The immanant generalizes both by introducing a character of the symmetric group $$S_n$$ in Leibniz's rule.

Determinants for finite-dimensional algebras
For any associative algebra $$A$$ that is finite-dimensional as a vector space over a field $$F$$, there is a determinant map
 * $$\det : A \to F.$$

This definition proceeds by establishing the characteristic polynomial independently of the determinant, and defining the determinant as the lowest order term of this polynomial. This general definition recovers the determinant for the matrix algebra $$A = \operatorname{Mat}_{n \times n}(F)$$, but also includes several further cases including the determinant of a quaternion,
 * $$\det (a + ib+jc+kd) = a^2 + b^2 + c^2 + d^2$$,

the norm $$N_{L/F} : L \to F$$ of a field extension, as well as the Pfaffian of a skew-symmetric matrix and the reduced norm of a central simple algebra, also arise as special cases of this construction.

Infinite matrices
For matrices with an infinite number of rows and columns, the above definitions of the determinant do not carry over directly. For example, in the Leibniz formula, an infinite sum (all of whose terms are infinite products) would have to be calculated. Functional analysis provides different extensions of the determinant for such infinite-dimensional situations, which however only work for particular kinds of operators.

The Fredholm determinant defines the determinant for operators known as trace class operators by an appropriate generalization of the formula
 * $$\det(I+A) = \exp(\operatorname{tr}(\log(I+A))). $$

Another infinite-dimensional notion of determinant is the functional determinant.

Operators in von Neumann algebras
For operators in a finite factor, one may define a positive real-valued determinant called the Fuglede−Kadison determinant using the canonical trace. In fact, corresponding to every tracial state on a von Neumann algebra there is a notion of Fuglede−Kadison determinant.

Related notions for non-commutative rings
For matrices over non-commutative rings, multilinearity and alternating properties are incompatible for $n ≥ 2$, so there is no good definition of the determinant in this setting.

For square matrices with entries in a non-commutative ring, there are various difficulties in defining determinants analogously to that for commutative rings. A meaning can be given to the Leibniz formula provided that the order for the product is specified, and similarly for other definitions of the determinant, but non-commutativity then leads to the loss of many fundamental properties of the determinant, such as the multiplicative property or that the determinant is unchanged under transposition of the matrix. Over non-commutative rings, there is no reasonable notion of a multilinear form (existence of a nonzero with a regular element of R as value on some pair of arguments implies that R is commutative). Nevertheless, various notions of non-commutative determinant have been formulated that preserve some of the properties of determinants, notably quasideterminants and the Dieudonné determinant. For some classes of matrices with non-commutative elements, one can define the determinant and prove linear algebra theorems that are very similar to their commutative analogs. Examples include the q-determinant on quantum groups, the Capelli determinant on Capelli matrices, and the Berezinian on supermatrices (i.e., matrices whose entries are elements of $$\mathbb Z_2$$-graded rings). Manin matrices form the class closest to matrices with commutative elements.

Calculation
Determinants are mainly used as a theoretical tool. They are rarely calculated explicitly in numerical linear algebra, where for applications such as checking invertibility and finding eigenvalues the determinant has largely been supplanted by other techniques. Computational geometry, however, does frequently use calculations related to determinants.

While the determinant can be computed directly using the Leibniz rule this approach is extremely inefficient for large matrices, since that formula requires calculating $$n!$$ ($$n$$ factorial) products for an $$n \times n$$-matrix. Thus, the number of required operations grows very quickly: it is of order $$n!$$. The Laplace expansion is similarly inefficient. Therefore, more involved techniques have been developed for calculating determinants.

Gaussian elemination
Gaussian elimination consists of left multiplying a matrix by elementary matrices for getting a matrix in a row echelon form. One can restrict the computation to elementary matrices of determinant $1$. In this case, the determinant of the resulting row echelon form equals the determinant of the initial matrix. As a row echelon form is a triangular matrix, its determinant is the product of the entries of its diagonal.

So, the determinant can be computed for almost free from the result of a Gaussian elemination.

Decomposition methods
Some methods compute $$\det(A)$$ by writing the matrix as a product of matrices whose determinants can be more easily computed. Such techniques are referred to as decomposition methods. Examples include the LU decomposition, the QR decomposition or the Cholesky decomposition (for positive definite matrices). These methods are of order $$\operatorname O(n^3)$$, which is a significant improvement over $$\operatorname O (n!)$$.

For example, LU decomposition expresses $$A$$ as a product
 * $$ A = PLU. $$

of a permutation matrix $$P$$ (which has exactly a single $$1$$ in each column, and otherwise zeros), a lower triangular matrix $$L$$ and an upper triangular matrix $$U$$. The determinants of the two triangular matrices $$L$$ and $$U$$ can be quickly calculated, since they are the products of the respective diagonal entries. The determinant of $$P$$ is just the sign $$\varepsilon$$ of the corresponding permutation (which is $$+1$$ for an even number of permutations and is $$ -1 $$ for an odd number of permutations). Once such a LU decomposition is known for $$A$$, its determinant is readily computed as

 * $$ \det(A) = \varepsilon \det(L)\cdot\det(U). $$

Further methods
The order $$\operatorname O(n^3)$$ reached by decomposition methods has been improved by different methods. If two matrices of order $$n$$ can be multiplied in time $$M(n)$$, where $$M(n) \ge n^a$$ for some $$a>2$$, then there is an algorithm computing the determinant in time $$O(M(n))$$. This means, for example, that an $$\operatorname O(n^{2.376})$$ algorithm for computing the determinant exists based on the Coppersmith–Winograd algorithm. This exponent has been further lowered, as of 2016, to 2.373.

In addition to the complexity of the algorithm, further criteria can be used to compare algorithms. Especially for applications concerning matrices over rings, algorithms that compute the determinant without any divisions exist. (By contrast, Gauss elimination requires divisions.) One such algorithm, having complexity $$\operatorname O(n^4)$$ is based on the following idea: one replaces permutations (as in the Leibniz rule) by so-called closed ordered walks, in which several items can be repeated. The resulting sum has more terms than in the Leibniz rule, but in the process several of these products can be reused, making it more efficient than naively computing with the Leibniz rule. Algorithms can also be assessed according to their bit complexity, i.e., how many bits of accuracy are needed to store intermediate values occurring in the computation. For example, the Gaussian elimination (or LU decomposition) method is of order $$\operatorname O(n^3)$$, but the bit length of intermediate values can become exponentially long. By comparison, the Bareiss Algorithm, is an exact-division method (so it does use division, but only in cases where these divisions can be performed without remainder) is of the same order, but the bit complexity is roughly the bit size of the original entries in the matrix times $$n$$.

If the determinant of A and the inverse of A have already been computed, the matrix determinant lemma allows rapid calculation of the determinant of $A + uv^{T}$, where u and v are column vectors.

Charles Dodgson (i.e. Lewis Carroll of Alice's Adventures in Wonderland fame) invented a method for computing determinants called Dodgson condensation. Unfortunately this interesting method does not always work in its original form.

title: Determinant


text:
Talk:Dactylic hexameter

Could someone add transliteration from the greek on the page? I know some greek letters, but not enough to read it, and I hate coming across something in another language I can't even pronounce when I'm reading. It breaks things up and makes it harder to analyze if I can't pronounce what I'm seeing. Bobbias (talk) 19:33, 20 December 2009 (UTC) - I hesitate to make statements about how common thy terminology is, but I can say that all of the noteworthy poets at least one prominent United States university use the terms "stressed" and "unstressed" rather than "long" and "short." I haven't thought of a way to add the other terms without making the entry unwieldy, so I'll leave it to your discretion. :-) --Koyaanis Qatsi - stressed & unstressed are right for English, but Greek and Latin poetry work somewhat differently -- though they both HAD syllable stress, the poetry is better thought of in terms of sung music with the lengths of notes.  Greek also had pitch, but that's a different matter and not perfectly understood, anyway.  So, think of long syllables as half notes and short syllables as quarter notes and you're not too far off - and it will help you remember that the poetry was mean to be performed orally even when it was composed on the page.  This system was somewhat artificial for Latin and had been adopted wholesale from Greek metrics, but it worked.  We should distinguish this on the entry. --MichaelTinkler - You're right; I was thinking only of English; and it's best to keep that distinction.  I have no experience with Greek or Latin poetry, aside from reading them in mediocre to good translations; I think I'll defer to you. --KQ - What about "heavy" and "light"? Some academics use these terms to avoid confusion with vowel length (long vowels [or diphthongs] make a syllable heavy but short ones don't necessarily make it light).

Also, can anyone vouch for the reasoning given for the ill repute of the Cicero line? I thought it had more to do with the cacophonous repetition of the syllables "natam" (and the unbelievably arrogant sentiment). Ou tis 23:00, 27 March 2006 (UTC) - I like the terms heavy and light. And I agree with you about Cicero. Overuse of spondees was bad, and their use in both of the first two feet in a line was generally frowned upon. But the heavily spondaic line was admissable and even preferable in certain circumstances - an author might wish, for example, to create a sort of spoken equivalent of a dramatic pause - and I think it's alright in this Cicero line provided he didn't do it too often (which he didn't). Cicero's metrical style is usually condemned as forced and repetitive, containing technically proficient lines but with little enjambment or interesting variation in word order, metre or placement of hiatus. In fact Cicero seems to have established many of the conventions used by later poets. For really bad lines look to Ennius where there are examples of almost everything that was avoided by the later poets.--Lo2u 15:30, 8 May 2006 (UTC)

is there any way we can hear these?
are there english and greek examples online, that could give foreigners a rough idea how they should sound?

Unicode notation?
Instead of using the clumsy use of "U" to denote two-shorts-or-one-long, we could use Unicode to get the proper signs, like this:

However, while this looks fine to me, I suspect some may lack the proper fonts and get squares or something else instead of the intended signs, so I'm a bit wary of making the edit. Are there guidelines for things like this? Alatius 20:02, 18 July 2007 (UTC)

I get squares, I imagine this is a common problem -- might be best to either go back to "U" or create an image of it.118.71.10.146 (talk) 06:21, 20 October 2009 (UTC)

Better Arrangement and Explanation
Several comments regarding the caesura were incorrect, and the section on the Greek was very thin. I thought I'd clean this article up to show the point of many of these rules regarding word placement, caesura, etc. and thereby underscore the counterpoint between accentual and metric rhythm which is a hallmark of dactylic hexameter.

In contrast to Homeric epic, Latin epic is highly rhetorical, and the meter plays an important role. Virgil is particularly important in this trend so I thought it was fair to expand this section and explain how the meter can be used to create verbal effects. I also just had to include the quote from Horace; though comical in its own right, it also shows how seriously these rules--which seem rather artificial in the modern age-- were taken by Roman writers.

The hexameter was also employed by late Latin and Medieval writers, sometimes with unusual results; I thought a mention of these variations was worthwhile (Latin poetry did not end with Virgil).

I also added some links to sound files I saw on-line. chjones_60656

Digamma
There are two places in Iliad I.108, I believe, which show the influence of the lost digamma. The first is at the third foot caesura, where there is now a hiatus. — Preceding unsigned comment added by Seadowns (talk • contribs) 10:47, 11 September 2017 (UTC)

Metamorphoses and De Rerum Natura
Ovid used this form in this famous work. Should it not be mentioned? - Tbsdy lives (talk) 12:37, 30 January 2008 (UTC)

Lucretius definitely also needs a mention.

Added Metamorphoses. 108.69.134.228 (talk) 09:46, 12 February 2016 (UTC)

Humanist Latin Poets
I think the article brushes aside the humanist poets too easily. There was a really remarkable output of hexameter verse by people such as Vida, Politian, Pontanus, Fracastor, Naugerius, Palearius, to name some Italians alone. If not the highest poetry, they showed an amazing virtuosity in their command of Latin. A good place to find them, recommended by Henry Hallam, is Pope's anthology Selecta Poemata Italorum Qui Latine Scripserunt. Seadowns (talk) 09:44, 2 September 2017 (UTC)

Bridges
Is the description of "Meyer's bridge" in the article correct?
 * The first, known as Meyer's Bridge, is in the second foot: if the second foot is a dactyl, the two short syllables must be part of the same word-unit.

As stated, this seems to be violated by the opening lines of the Iliad and the Aeneid both:
 * μῆνιν ἄειδε, θεά, Πηληϊάδεω Ἀχιλῆος
 * Arma virumque cano, Troiae qui primus ab oris

As far as I understand the quoted rule, it says that the word break between "ἄειδε" and "θεά" is not allowed, nor is the one between "virumque" and "cano". Neither does the English verse
 * This is the forest primeval. The murmuring pines and the hemlocks

follow the rule. If it is as universal as it is implied, shouldn't the article contain more examples that actually respect it? –Henning Makholm (talk) 15:28, 15 November 2009 (UTC)
 * This stuff about bridges should be left out. Any value it may have would be too advanced for a rudimentary article like this. Also, is a word-unit something different from a word? Seadowns (talk) 00:37, 2 September 2017 (UTC)
 * I agree. It isn't an essential part of the description of a hexameter, in view of the fact that the very first line of the Iliad contradicts it. I will delete the paragraph. Kanjuzi (talk) 06:02, 3 September 2017 (UTC)

Caesuras
The statement that a caesura is a break in sense, rather like a comma, is incorrect. It is the end of a word, which may or may not coincide with a break in sense. Seadowns (talk) 18:42, 28 August 2017 (UTC)
 * Quite right. I have changed it. Kanjuzi (talk) 05:21, 30 August 2017 (UT

I suggest cutting out the last two sentences of the paragraph.. The last but one is unimportant, if true. The last one has no meaning that I can find. Is there an example of what it is trying to say? Seadowns (talk) 00:29, 2 September 2017 (UTC)

The line from Horace, without a proper caesura in the third or fourth foot, follows a permissible pattern, if I remember correctly. In this pattern the second foot has to be a dactyl, with a break after the long and the two shorts in one word, and, I had thought, it had to end in an elision, though this one doesn't. Composers ought to stick to the elision I think. Here is an example from a modern composer -"cingebant nemora herbosos in sole recessus", showing the elision. Seadowns (talk) 13:43, 11 April 2021 (UTC)

Grammar First, then Poetry
"The sixth foot is always a spondee, though it may be anceps". "Always" means forever, permanently... there should be no "though" as "always" is an absolute; never varied. Please fix it whom is better then I at poetry. 108.38.36.17 (talk) 23:16, 26 February 2012 (UTC)

Bolded note?
I have no idea why the second note on this page is written in bold text. I've tried looking at the page but can't seem to figure it out. Anybody else know? 23:12, 23 October 2012 (UTC) — Preceding unsigned comment added by 50.4.35.41 (talk)

English hexameter
English metrical form relies on stress, not quantity. The vowel sound in 'pig' is not made longer by having two consonants after it. Pamour (talk) 21:47, 17 March 2013 (UTC)

Well, this one does say it's in quantitative meter, and I think it does a good job at demonstrating how that would feel in English. But you are right, there is a piglet in the ointment. To fix that, you could pronounce the g syllabically (or as guh), start the m sound early (pigm-), insert an a- prefix (a-munching) - or perhaps replace the pig with a boar. But I do like it the way it is. Perhaps the word break gives it some brevis in longo allowance? 81.159.72.169 (talk) 17:13, 5 August 2013 (UTC)

The deep dark dell line can be read as in stress metre too. It is the first line of an elegiac couplet, and the pentameter reads "Out of her mouth came forth yesterday's dinner and tea". I don't know the poet's name. Seadowns (talk) 10:53, 11 September 2017 (UTC)

Ennius's spondees
"The following lines of Ennius would not have been felt admissible by later authors since they both contain repeated spondees at the beginning of consecutive lines..." But this isn't actually true, since for example Virgil writes in Dido's curse, in consecutive lines (Aeneid 4.607f):
 * Sol, qui terrarum flammis opera omnia lustras,
 * – – | – – | – – | – u u | – u u | – – |
 * tuque harum interpres curarum et conscia Iuno,
 * – – | – – | – – | – – | – u u | – – |

Perhaps the statement should be modified or omitted therefore. Kanjuzi (talk) 20:40, 29 March 2017 (UTC)

WRONG
NOT too hard to make "quantitative"

''Heartless soulless brainless shameless needless all elders

With those firearms* from hell (madnesses under utter lies).

''What then now? when then that next death by gunner-elders?''

''Oh what an evil! utter dystopia of all-gunner earth-world.''

Youths many ladies as well: who then victim anew now?

Guiltless dead now, underground by hand number of wrong.

''Mighty police**? number all with a gun-thing a swift unaliveness.''

''Where then law? where law? here not not much thereof indeed,''

Hatred alone many imperialist steady shameless utter lies.

 * That's very good! Although surely underground can't be scanned – – – or utter lies u – –, unless perhaps you have a very strong Irish accent? Kanjuzi (talk) 10:27, 10 May 2022 (UTC)

"dactylic" is a misnomer
If you look at the teachings by the ancients on the heroics/hexameters, you'll find that the dactyl was not at the heart of it. Why? Because long syllables add solemnity, not short syllables. The short syllables simply added variation, levity, flexibility. The hexameters were meant to be solemn, such that the more spondees, the better/more serious. — Preceding unsigned comment added by 73.57.144.205 (talk • contribs) 19:36, 4 December 2017 (UTC)

 * By that measure vergil is a better poet than Homer. 2600:8806:340B:E100:A0EA:A3B4:8CB3:563A 03:19, 2 March 2021 (UTC)

Change the divisions between feet to a '/' ?
Specifically on the Latin section, the divisions between the feet look awfully like an 'l'. Perhaps using a '/' or breaks of some sort would make it less confusing, especially to those learning about this topic for the first time. PeelitePowellite (talk) 21:38, 30 May 2019 (UTC)

Text formatting?
I've spent some time reading this page on my Android phone with no problems whatever, but after switching to my laptop, which is running Ubuntu 18, I've found that although the long syllable sign and anceps signs are perfectly legible, I can no longer see the short syllable sign (as you all know, the bottom half of a circle). I have this problem both in Brave, my browser of choice, and in Firefox.

I only point this out, because on other articles where potential formatting problems are present, information is provided in a box at the top indicating which upgrades the user needs. I'm too backward with tech to be able to do this for myself, so I leave the matter in your capable hands.

Daedalus 96 (talk) 13:56, 15 September 2020 (UTC)

 * That's very interesting. Are you talking about the short syllable sign in the general pattern (–⏕ | –⏕ | –⏕ | –⏕ | –⏑⏑ | –X), or as it appears on top of letters such as ǎ? Kanjuzi (talk) 06:46, 9 May 2022 (UTC)

Character of Hexameters
I think the article gives too narrow a view of hexameters, in Latin at any rate. What it says about their epic character is quite true, but they were also used for didactic, pastoral and satirical poetry, and for lighter, conversational poems like Horace's Epistles. They were a sort of poetic maid-of-all-work. If people agree perhaps the article could be filled out to cover this. Someone may know more about the Greek position than I do, but at least I know Hesiod and the pastorals used them. Seadowns (talk) 14:10, 10 April 2021 (UTC)

"Heavily Spondaic Line"
Hi team, the article currently has this section regarding the change over time from a spondee preference to a dactylic preference for the first four feet:

''However, it is from Virgil that the following famous, heavily spondaic line comes:

''mōnstrum hor|rendum, īn|fōrme, in|gēns, cui| lūmen a|demptum. (Aeneid III.658) "a huge, shapeless, horrendous monster, whose light [i.e., its eye] had been removed"''

However, this line is heavily dactylic, not heavily spondaic. The scansion would be DDDS[DS], wouldn't it? Is this a 'brain fart' on the part of the author, or am I missing something about the gradual shift from spondee preference to dactyl preference for the first four feet? This line is meant to contrast with the lines:

''hīs ver|bīs: ō| gnāta, ti|bī sunt| ante fe|rendae aerum|nae, post| ex fluvi|ō for|tūna re|sistet. (Annales 1.42f)''

which are indeed heavily spondaic: SSDSDS and SSSSDS, respectively.

All this is to say, should the line

However, it is from Virgil that the following famous, heavily spondaic line comes:

instead say:

However, it is from Virgil that the following famous, heavily datylic line comes:

(and, this being the case, is it really the most dactylic line in the Aeneid? Surely there's gotta be a DDDDDS line somewhere in the heavy action of the later books?) — Preceding unsigned comment added by Mccartneyac (talk • contribs) 13:56, 7 May 2021 (UTC)

 * It is heavily spondiac. I origninally added this because the text said "too many spondees were inadmissible by the time of Vergil", whereas the line really is famous (more so than Ennius to us moderns) and contains no less spondees than "hīs ver|bīs: "ō| gnāta, ti|bī sunt| ante fe|rendae". So, the intention is: "Many spondees might indeed have become less popular, but there are exceptions, and the most famous line with many spondees actually is from Vergil".
 * "Am I missing something?" Yes, the elision of hiatuses. - The problem is that there is a tradition not to represent the elision of hiatuses in writing (though I think I remember I originally did that), and then people might overlook that these things are there.
 * When one reads the thing, one reads
 * monstr' horr|end' in|form' in|gens cui |lumen ad|emptum,
 * (I leave the natural lengths out because the lengths would, in any case, be positional), so it should be quite clear that the line is heavily spondiac.--138.245.1.1 (talk) 12:58, 28 July 2021 (UTC)

MOS DASH etc.
. (a) Concerning MOS:DASH, the recommendation there is to use spaces round an N-dash, and no spaces round an M-dash. These, as far as I can see, are only N-dashes, which is why I added spaces.

(b) It seems to me that the correct place to put the foot separation with instances like quati/t ungula is before the t, since the t is taken metrically as part of the following syllable.

(c) I can see no objection to mentioning in the lead that a hexameter usually has a caesura in the 3rd foot. It does seem an important and typical feature of most hexameters, so it should be included as basic information. The variations, such as 4th-foot caesura or 5th-foot spondee, can be mentioned in the main text. Kanjuzi (talk) 18:17, 7 May 2022 (UTC)
 * (a) No, they're em dashes. The article has unspaced em dashes rather than spaced en dashes everywhere throughout.
 * (b) How is the t "taken metrically as part of the following syllable"? The syllable would be "short" whether it were "tit" or "ti", and adding t to the next syllable doesn't change its quantity. The divisions should follow the natural syllabication of the words.
 * (c) The problem was that "either after the first, or after the second syllable of the 3rd foot" is awkward syntactically. You didn't like my initial rewording of the sentence, so I tried just removing the sentence as superfluous. "Usually there is a break, called a caesura, after either the first or the second syllable of the third foot" would be syntactically OK, but it obscures the fact that caesurae can be in other positions. Deor (talk) 18:58, 7 May 2022 (UTC)
 * (a) Concerning the M-dash, I see that I was mistaken. On the edit page, the N dash and M dash are identical, but once it's published the M-dash looks longer.
 * (b) No, that's not correct. For the fact is that the last syllable of any word, spoken in isolation or at the end of a verse or sentence, is long. It only becomes short when another word immediately follows. That is why a word like condidit or dicere can very well stand at the end of an iambic line (u – u –); but it cannot come at the end of a hexameter, since if it did it would automatically turn into a cretic (– u –).
 * (c) Since Virgil has a masculine caesura in more than 85% of his lines I think it would be fair to say that there is usually a word-break in the middle of the third foot. Lines which have no 3rd foot caesura at all, such as haec secum: ‘mene incepto desistere victam are very rare. It does seem to be an important part of the general description, qualified by the word "usually". It doesn't make any difference that it is discussed in greater detail in the body of the article, since you have to remember that a lot of users just look at the lead and don't read any further; so all the basic information has to be there. Kanjuzi (talk) 06:42, 9 May 2022 (UTC)
 * Re (b): I don't see the relevance of your analysis. Clearly quatit isn't "at the end of a verse or sentence", since it's immediately followed by ungula in that line. Only a syllable can be long or short in scansion, and the second syllable in quatit is "tit", not "ti". I don't seem to have at hand an example of reliable source containing a scanned line to illustrate this, but there are several in Prosody (Latin) (see, for example, the scansion of the last two feet of Aeneid 1.3 here). Re (c): You can restore the caesura sentence in the lead if you want, but I think it would be advisable to use the revision I suggested above to avoid faulty parallelism in the sentence. Deor (talk) 20:28, 9 May 2022 (UTC)
 * I agree with you that quatit has two short syllables when it is followed by ungula; but that's only because the final consonant in pronunciation becomes part of the following syllable. So quatit ungula would be syllabified qua-ti-tun-gu-la. Since it's syllabified that way, I think you should consider the 5th foot to be /tungula/. It's like when you say las arbres (the trees) in French: it's pronounced lézarbr and you would syllabify it and pronounce it as lé-zarbr not léz-arbr. Kanjuzi (talk) 10:15, 10 May 2022 (UTC)
 * I think we're talking past each other to some extent here, and I'm getting weary of the whole matter. You can syllabicate the Latin in whatever way seems best to you, and I won't revert. I'll probably just remove the article from my watchlist. Deor (talk) 13:43, 10 May 2022 (UTC)

Spondaic lines
I draw attention to the sentence "The following lines of Ennius would not have been felt admissible by later authors since they both contain repeated spondees at the beginning of consecutive lines...". Where does this idea come from? It is certainly incorrect. In Aeneid 1, for example, we find lines beginning with two spondees at 15, 31, 32, 34, 36, 37 , 40, 44...., so it is neither rare nor forbidden for it to occur in consecutive lines. It would really be better if this article were based on standard published works on Greek and Latin metre, none of which seems to have been quoted as yet. Kanjuzi (talk) 08:36, 9 May 2022 (UTC)

Learning from rules?
In the current version of the article: "The hexameter came into Latin as an adaptation from Greek long after the practice of singing the epics had faded. Consequentially, the properties of the meter were learned as specific "rules" rather than as a natural result of musical expression." There is surely no evidence for this. Mentions of the hexameter in metrical writers are not very detailed and don't discuss specific rules at all. Also it seems unlikely. Most people, when composing poetry in their own language, simply write what comes naturally in imitation of patterns they have heard before; they don't consult a rule-book or need lessons. So I think this idea also needs to go. Kanjuzi (talk) 10:57, 10 May 2022 (UTC)

Syllables
As you can see, I have added a paragraph on how to tell short from long syllables. I hope this will not seem unnecessary to anyone. First of all, the article on Scansion is purely about English scansion so not helpful here. Secondly, this metre is the probably the first or even the only metre students of Latin will study, and knowing syllable lengths is a necessary preliminary.

Another thought concerning this article is this. Many readers will be interested only in Latin, since the number who study Ancient Greek as well is very limited. So they are likely to skip over any paragraphs discussing Greek. For that reason, it might be better to do the introductory paragraph on how to scan a line into feet using a Latin example, rather than a Greek one, and then after that to add a paragraph on Greek hexameter poetry which wouldn't be essential reading for an understanding of the subject. Kanjuzi (talk) 16:52, 11 May 2022 (UTC)

Italian hexameter
It is totally absent a complete discussion about italian hexameter, both in original poetry and in traslation of classical poetry, as an exhaustive treatment of usage of hexameter in spanish, catalan and portuguese literature. 94.185.74.172 (talk) 10:05, 18 March 2024 (UTC)

title: Talk:Dactylic hexameter