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Orthogonal matrix

In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.

One way to express this is $$Q^\mathrm{T} Q = Q Q^\mathrm{T} = I,$$ where $Q^{T}$ is the transpose of $Q$ and $I$ is the identity matrix.

This leads to the equivalent characterization: a matrix $Q$ is orthogonal if its transpose is equal to its inverse: $$Q^\mathrm{T}=Q^{-1},$$ where $Q^{−1}$ is the inverse of $Q$.

An orthogonal matrix $Q$ is necessarily invertible (with inverse $Q^{−1} = Q^{T}$), unitary ($Q^{−1} = Q^{∗}$), where $Q^{∗}$ is the Hermitian adjoint (conjugate transpose) of $Q$, and therefore normal ($Q^{∗}Q = QQ^{∗}$) over the real numbers. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix preserves the inner product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation, reflection or rotoreflection. In other words, it is a unitary transformation.

The set of $n × n$ orthogonal matrices, under multiplication, forms the group $O(n)$, known as the orthogonal group. The subgroup $SO(n)$ consisting of orthogonal matrices with determinant +1 is called the special orthogonal group, and each of its elements is a special orthogonal matrix. As a linear transformation, every special orthogonal matrix acts as a rotation.

Overview

An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. Although we consider only real matrices here, the definition can be used for matrices with entries from any field. However, orthogonal matrices arise naturally from dot products, and for matrices of complex numbers that leads instead to the unitary requirement. Orthogonal matrices preserve the dot product, so, for vectors $u$ and $v$ in an $n$-dimensional real Euclidean space $${\mathbf u} \cdot {\mathbf v} = \left(Q {\mathbf u}\right) \cdot \left(Q {\mathbf v}\right) $$ where $Q$ is an orthogonal matrix. To see the inner product connection, consider a vector $v$ in an $n$-dimensional real Euclidean space. Written with respect to an orthonormal basis, the squared length of $v$ is $v^{T}v$. If a linear transformation, in matrix form $Qv$, preserves vector lengths, then $${\mathbf v}^\mathrm{T}{\mathbf v} = (Q{\mathbf v})^\mathrm{T}(Q{\mathbf v}) = {\mathbf v}^\mathrm{T} Q^\mathrm{T} Q {\mathbf v} .$$

Thus finite-dimensional linear isometries—rotations, reflections, and their combinations—produce orthogonal matrices. The converse is also true: orthogonal matrices imply orthogonal transformations. However, linear algebra includes orthogonal transformations between spaces which may be neither finite-dimensional nor of the same dimension, and these have no orthogonal matrix equivalent.

Orthogonal matrices are important for a number of reasons, both theoretical and practical. The $n × n$ orthogonal matrices form a group under matrix multiplication, the orthogonal group denoted by $O(n)$, which—with its subgroups—is widely used in mathematics and the physical sciences. For example, the point group of a molecule is a subgroup of O(3). Because floating point versions of orthogonal matrices have advantageous properties, they are key to many algorithms in numerical linear algebra, such as $QR$ decomposition. As another example, with appropriate normalization the discrete cosine transform (used in MP3 compression) is represented by an orthogonal matrix.

Examples
Below are a few examples of small orthogonal matrices and possible interpretations. \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix}$$    (identity transformation) \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{bmatrix}$$    (rotation about the origin) \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix}$$    (reflection across x-axis) \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix}$$    (permutation of coordinate axes)

Lower dimensions
The simplest orthogonal matrices are the 1 × 1 matrices [1] and [−1], which we can interpret as the identity and a reflection of the real line across the origin.

The 2 × 2 matrices have the form $$\begin{bmatrix} p & t\\ q & u \end{bmatrix},$$ which orthogonality demands satisfy the three equations $$\begin{align} 1 & = p^2+t^2, \\ 1 & = q^2+u^2, \\ 0 & = pq+tu. \end{align}$$

In consideration of the first equation, without loss of generality let $p = cos θ$, $q = sin θ$; then either $t = −q$, $u = p$ or $t = q$, $u = −p$. We can interpret the first case as a rotation by $θ$ (where $θ = 0$ is the identity), and the second as a reflection across a line at an angle of $θ⁄2$.

$$ \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{bmatrix}\text{ (rotation), }\qquad \begin{bmatrix} \cos \theta & \sin \theta \\ \sin \theta & -\cos \theta \\ \end{bmatrix}\text{ (reflection)} $$

The special case of the reflection matrix with $θ = 90°$ generates a reflection about the line at 45° given by $y = x$ and therefore exchanges $x$ and $y$; it is a permutation matrix, with a single 1 in each column and row (and otherwise 0): $$\begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}.$$

The identity is also a permutation matrix.

A reflection is its own inverse, which implies that a reflection matrix is symmetric (equal to its transpose) as well as orthogonal. The product of two rotation matrices is a rotation matrix, and the product of two reflection matrices is also a rotation matrix.

Higher dimensions
Regardless of the dimension, it is always possible to classify orthogonal matrices as purely rotational or not, but for 3 × 3 matrices and larger the non-rotational matrices can be more complicated than reflections. For example, $$ \begin{bmatrix} -1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & -1 \end{bmatrix}\text{ and } \begin{bmatrix} 0 & -1 & 0\\ 1 & 0 & 0\\ 0 & 0 & -1 \end{bmatrix}$$

represent an inversion through the origin and a rotoinversion, respectively, about the $z$-axis.

Rotations become more complicated in higher dimensions; they can no longer be completely characterized by one angle, and may affect more than one planar subspace. It is common to describe a 3 × 3 rotation matrix in terms of an axis and angle, but this only works in three dimensions. Above three dimensions two or more angles are needed, each associated with a plane of rotation.

However, we have elementary building blocks for permutations, reflections, and rotations that apply in general.

Primitives
The most elementary permutation is a transposition, obtained from the identity matrix by exchanging two rows. Any $n × n$ permutation matrix can be constructed as a product of no more than $n − 1$ transpositions.

A Householder reflection is constructed from a non-null vector $v$ as $$Q = I - 2 \frac{{\mathbf v}{\mathbf v}^\mathrm{T}}{{\mathbf v}^\mathrm{T}{\mathbf v}} .$$

Here the numerator is a symmetric matrix while the denominator is a number, the squared magnitude of $v$. This is a reflection in the hyperplane perpendicular to $v$ (negating any vector component parallel to $v$). If $v$ is a unit vector, then $Q = I − 2vv^{T}$ suffices. A Householder reflection is typically used to simultaneously zero the lower part of a column. Any orthogonal matrix of size n × n can be constructed as a product of at most $n$ such reflections.

A Givens rotation acts on a two-dimensional (planar) subspace spanned by two coordinate axes, rotating by a chosen angle. It is typically used to zero a single subdiagonal entry. Any rotation matrix of size $n × n$ can be constructed as a product of at most $n(n − 1)⁄2$ such rotations. In the case of 3 × 3 matrices, three such rotations suffice; and by fixing the sequence we can thus describe all 3 × 3 rotation matrices (though not uniquely) in terms of the three angles used, often called Euler angles.

A Jacobi rotation has the same form as a Givens rotation, but is used to zero both off-diagonal entries of a 2 × 2 symmetric submatrix.

Matrix properties
A real square matrix is orthogonal if and only if its columns form an orthonormal basis of the Euclidean space $R^{n}$ with the ordinary Euclidean dot product, which is the case if and only if its rows form an orthonormal basis of $R^{n}$. It might be tempting to suppose a matrix with orthogonal (not orthonormal) columns would be called an orthogonal matrix, but such matrices have no special interest and no special name; they only satisfy $M^{T}M = D$, with $D$ a diagonal matrix.

The determinant of any orthogonal matrix is +1 or −1. This follows from basic facts about determinants, as follows: $$1=\det(I)=\det\left(Q^\mathrm{T}Q\right)=\det\left(Q^\mathrm{T}\right)\det(Q)=\bigl(\det(Q)\bigr)^2 .$$

The converse is not true; having a determinant of ±1 is no guarantee of orthogonality, even with orthogonal columns, as shown by the following counterexample. $$\begin{bmatrix} 2 & 0 \\ 0 & \frac{1}{2} \end{bmatrix}$$

With permutation matrices the determinant matches the signature, being +1 or −1 as the parity of the permutation is even or odd, for the determinant is an alternating function of the rows.

Stronger than the determinant restriction is the fact that an orthogonal matrix can always be diagonalized over the complex numbers to exhibit a full set of eigenvalues, all of which must have (complex) modulus 1.

Group properties
The inverse of every orthogonal matrix is again orthogonal, as is the matrix product of two orthogonal matrices. In fact, the set of all $n × n$ orthogonal matrices satisfies all the axioms of a group. It is a compact Lie group of dimension $n(n − 1)⁄2$, called the orthogonal group and denoted by $O(n)$.

The orthogonal matrices whose determinant is +1 form a path-connected normal subgroup of $O(n)$ of index 2, the special orthogonal group $SO(n)$ of rotations. The quotient group $O(n)/SO(n)$ is isomorphic to $O(1)$, with the projection map choosing [+1] or [−1] according to the determinant. Orthogonal matrices with determinant −1 do not include the identity, and so do not form a subgroup but only a coset; it is also (separately) connected. Thus each orthogonal group falls into two pieces; and because the projection map splits, $O(n)$ is a semidirect product of $SO(n)$ by $O(1)$. In practical terms, a comparable statement is that any orthogonal matrix can be produced by taking a rotation matrix and possibly negating one of its columns, as we saw with 2 × 2 matrices. If $n$ is odd, then the semidirect product is in fact a direct product, and any orthogonal matrix can be produced by taking a rotation matrix and possibly negating all of its columns. This follows from the property of determinants that negating a column negates the determinant, and thus negating an odd (but not even) number of columns negates the determinant.

Now consider $(n + 1) × (n + 1)$ orthogonal matrices with bottom right entry equal to 1. The remainder of the last column (and last row) must be zeros, and the product of any two such matrices has the same form. The rest of the matrix is an $n × n$ orthogonal matrix; thus $O(n)$ is a subgroup of $O(n + 1)$ (and of all higher groups).

$$\begin{bmatrix} & & & 0\\ & \mathrm{O}(n) & & \vdots\\ & & & 0\\ 0 & \cdots & 0 & 1 \end{bmatrix}$$

Since an elementary reflection in the form of a Householder matrix can reduce any orthogonal matrix to this constrained form, a series of such reflections can bring any orthogonal matrix to the identity; thus an orthogonal group is a reflection group. The last column can be fixed to any unit vector, and each choice gives a different copy of $O(n)$ in $O(n + 1)$; in this way $O(n + 1)$ is a bundle over the unit sphere $S^{n}$ with fiber $O(n)$.

Similarly, $SO(n)$ is a subgroup of $SO(n + 1)$; and any special orthogonal matrix can be generated by Givens plane rotations using an analogous procedure. The bundle structure persists: $SO(n) ↪ SO(n + 1) → S^{n}$. A single rotation can produce a zero in the first row of the last column, and series of $n − 1$ rotations will zero all but the last row of the last column of an $n × n$ rotation matrix. Since the planes are fixed, each rotation has only one degree of freedom, its angle. By induction, $SO(n)$ therefore has $$(n-1) + (n-2) + \cdots + 1 = \frac{n(n-1)}{2}$$ degrees of freedom, and so does $O(n)$.

Permutation matrices are simpler still; they form, not a Lie group, but only a finite group, the order $n!$ symmetric group $S_{n}$. By the same kind of argument, $S_{n}$ is a subgroup of $S_{n + 1}$. The even permutations produce the subgroup of permutation matrices of determinant +1, the order $n!⁄2$ alternating group.

Canonical form
More broadly, the effect of any orthogonal matrix separates into independent actions on orthogonal two-dimensional subspaces. That is, if $Q$ is special orthogonal then one can always find an orthogonal matrix $P$, a (rotational) change of basis, that brings $Q$ into block diagonal form:

$$P^\mathrm{T}QP = \begin{bmatrix} R_1 & & \\ & \ddots & \\ & & R_k \end{bmatrix}\ (n\text{ even}), \ P^\mathrm{T}QP = \begin{bmatrix} R_1 & & & \\ & \ddots & & \\ & & R_k & \\ & & & 1 \end{bmatrix}\ (n\text{ odd}).$$

where the matrices $R_{1}, ..., R_{k}$ are 2 × 2 rotation matrices, and with the remaining entries zero. Exceptionally, a rotation block may be diagonal, $±I$. Thus, negating one column if necessary, and noting that a 2 × 2 reflection diagonalizes to a +1 and −1, any orthogonal matrix can be brought to the form $$P^\mathrm{T}QP = \begin{bmatrix} \begin{matrix}R_1 & & \\ & \ddots & \\ & & R_k\end{matrix} & 0 \\ 0 & \begin{matrix}\pm 1 & & \\ & \ddots & \\ & & \pm 1\end{matrix} \\ \end{bmatrix},$$

The matrices $R_{1}, ..., R_{k}$ give conjugate pairs of eigenvalues lying on the unit circle in the complex plane; so this decomposition confirms that all eigenvalues have absolute value 1. If $n$ is odd, there is at least one real eigenvalue, +1 or −1; for a 3 × 3 rotation, the eigenvector associated with +1 is the rotation axis.

Lie algebra
Suppose the entries of $Q$ are differentiable functions of $t$, and that $t = 0$ gives $Q = I$. Differentiating the orthogonality condition $$Q^\mathrm{T} Q = I $$ yields $$\dot{Q}^\mathrm{T} Q + Q^\mathrm{T} \dot{Q} = 0$$

Evaluation at $t = 0$ ($Q = I$) then implies $$\dot{Q}^\mathrm{T} = -\dot{Q} .$$

In Lie group terms, this means that the Lie algebra of an orthogonal matrix group consists of skew-symmetric matrices. Going the other direction, the matrix exponential of any skew-symmetric matrix is an orthogonal matrix (in fact, special orthogonal).

For example, the three-dimensional object physics calls angular velocity is a differential rotation, thus a vector in the Lie algebra $$\mathfrak{so}(3)$$ tangent to $SO(3)$. Given $ω = (xθ, yθ, zθ)$, with $v = (x, y, z)$ being a unit vector, the correct skew-symmetric matrix form of $ω$ is $$ \Omega = \begin{bmatrix} 0 & -z\theta & y\theta \\ z\theta & 0 & -x\theta \\ -y\theta & x\theta & 0 \end{bmatrix} .$$

The exponential of this is the orthogonal matrix for rotation around axis $v$ by angle $θ$; setting $c = cos θ⁄2$, $s = sin θ⁄2$, $$\exp(\Omega) = \begin{bmatrix} 1 -  2s^2  +  2x^2 s^2  &  2xy s^2  -  2z sc  &  2xz s^2  +  2y sc\\ 2xy s^2 +  2z sc  &  1  -  2s^2  +  2y^2 s^2  &  2yz s^2  -  2x sc\\ 2xz s^2 -  2y sc  &  2yz s^2  +  2x sc  &  1  -  2s^2  +  2z^2 s^2 \end{bmatrix}.$$

Benefits
Numerical analysis takes advantage of many of the properties of orthogonal matrices for numerical linear algebra, and they arise naturally. For example, it is often desirable to compute an orthonormal basis for a space, or an orthogonal change of bases; both take the form of orthogonal matrices. Having determinant ±1 and all eigenvalues of magnitude 1 is of great benefit for numeric stability. One implication is that the condition number is 1 (which is the minimum), so errors are not magnified when multiplying with an orthogonal matrix. Many algorithms use orthogonal matrices like Householder reflections and Givens rotations for this reason. It is also helpful that, not only is an orthogonal matrix invertible, but its inverse is available essentially free, by exchanging indices.

Permutations are essential to the success of many algorithms, including the workhorse Gaussian elimination with partial pivoting (where permutations do the pivoting). However, they rarely appear explicitly as matrices; their special form allows more efficient representation, such as a list of $n$ indices.

Likewise, algorithms using Householder and Givens matrices typically use specialized methods of multiplication and storage. For example, a Givens rotation affects only two rows of a matrix it multiplies, changing a full multiplication of order $n^{3}$ to a much more efficient order $n$. When uses of these reflections and rotations introduce zeros in a matrix, the space vacated is enough to store sufficient data to reproduce the transform, and to do so robustly. (Following, we do not store a rotation angle, which is both expensive and badly behaved.)

Decompositions
A number of important matrix decompositions involve orthogonal matrices, including especially:

 * $QR$ decomposition : $M = QR$, $Q$ orthogonal, $R$ upper triangular
 * Singular value decomposition : $M = UΣV^{T}$, $U$ and $V$ orthogonal, $Σ$ diagonal matrix
 * Eigendecomposition of a symmetric matrix (decomposition according to the spectral theorem) : $S = QΛQ^{T}$, $S$ symmetric, $Q$ orthogonal, $Λ$ diagonal
 * Polar decomposition : $M = QS$, $Q$ orthogonal, $S$ symmetric positive-semidefinite

Examples
Consider an overdetermined system of linear equations, as might occur with repeated measurements of a physical phenomenon to compensate for experimental errors. Write $Ax = b$, where $A$ is $m × n$, $m > n$. A $QR$ decomposition reduces $A$ to upper triangular $R$. For example, if $A$ is 5 × 3 then $R$ has the form $$R = \begin{bmatrix} \cdot & \cdot & \cdot \\ 0 & \cdot & \cdot \\ 0 & 0 & \cdot \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}.$$

The linear least squares problem is to find the $x$ that minimizes $\|Ax − b\|$, which is equivalent to projecting $b$ to the subspace spanned by the columns of $A$. Assuming the columns of $A$ (and hence $R$) are independent, the projection solution is found from $A^{T}Ax = A^{T}b$. Now $A^{T}A$ is square ($n × n$) and invertible, and also equal to $R^{T}R$. But the lower rows of zeros in $R$ are superfluous in the product, which is thus already in lower-triangular upper-triangular factored form, as in Gaussian elimination (Cholesky decomposition). Here orthogonality is important not only for reducing $A^{T}A = (R^{T}Q^{T})QR$ to $R^{T}R$, but also for allowing solution without magnifying numerical problems.

In the case of a linear system which is underdetermined, or an otherwise non-invertible matrix, singular value decomposition (SVD) is equally useful. With $A$ factored as $UΣV^{T}$, a satisfactory solution uses the Moore-Penrose pseudoinverse, $VΣ^{+}U^{T}$, where $Σ^{+}$ merely replaces each non-zero diagonal entry with its reciprocal. Set $x$ to $VΣ^{+}U^{T}b$.

The case of a square invertible matrix also holds interest. Suppose, for example, that $A$ is a 3 × 3 rotation matrix which has been computed as the composition of numerous twists and turns. Floating point does not match the mathematical ideal of real numbers, so $A$ has gradually lost its true orthogonality. A Gram–Schmidt process could orthogonalize the columns, but it is not the most reliable, nor the most efficient, nor the most invariant method. The polar decomposition factors a matrix into a pair, one of which is the unique closest orthogonal matrix to the given matrix, or one of the closest if the given matrix is singular. (Closeness can be measured by any matrix norm invariant under an orthogonal change of basis, such as the spectral norm or the Frobenius norm.) For a near-orthogonal matrix, rapid convergence to the orthogonal factor can be achieved by a "Newton's method" approach due to (1990), repeatedly averaging the matrix with its inverse transpose. has published an accelerated method with a convenient convergence test.

For example, consider a non-orthogonal matrix for which the simple averaging algorithm takes seven steps $$\begin{bmatrix}3 & 1\\7 & 5\end{bmatrix} \rightarrow \begin{bmatrix}1.8125 & 0.0625\\3.4375 & 2.6875\end{bmatrix} \rightarrow \cdots \rightarrow \begin{bmatrix}0.8 & -0.6\\0.6 & 0.8\end{bmatrix}$$ and which acceleration trims to two steps (with $γ$ = 0.353553, 0.565685).

$$\begin{bmatrix}3 & 1\\7 & 5\end{bmatrix} \rightarrow \begin{bmatrix}1.41421 & -1.06066\\1.06066 & 1.41421\end{bmatrix} \rightarrow \begin{bmatrix}0.8 & -0.6\\0.6 & 0.8\end{bmatrix}$$

Gram-Schmidt yields an inferior solution, shown by a Frobenius distance of 8.28659 instead of the minimum 8.12404.

$$\begin{bmatrix}3 & 1\\7 & 5\end{bmatrix} \rightarrow \begin{bmatrix}0.393919 & -0.919145\\0.919145 & 0.393919\end{bmatrix}$$

Randomization
Some numerical applications, such as Monte Carlo methods and exploration of high-dimensional data spaces, require generation of uniformly distributed random orthogonal matrices. In this context, "uniform" is defined in terms of Haar measure, which essentially requires that the distribution not change if multiplied by any freely chosen orthogonal matrix. Orthogonalizing matrices with independent uniformly distributed random entries does not result in uniformly distributed orthogonal matrices, but the $QR$ decomposition of independent normally distributed random entries does, as long as the diagonal of $R$ contains only positive entries. replaced this with a more efficient idea that later generalized as the "subgroup algorithm" (in which form it works just as well for permutations and rotations). To generate an $(n + 1) × (n + 1)$ orthogonal matrix, take an $n × n$ one and a uniformly distributed unit vector of dimension n + 1. Construct a Householder reflection from the vector, then apply it to the smaller matrix (embedded in the larger size with a 1 at the bottom right corner).

Nearest orthogonal matrix
The problem of finding the orthogonal matrix $Q$ nearest a given matrix $M$ is related to the Orthogonal Procrustes problem. There are several different ways to get the unique solution, the simplest of which is taking the singular value decomposition of $M$ and replacing the singular values with ones. Another method expresses the $R$ explicitly but requires the use of a matrix square root: $$Q = M \left(M^\mathrm{T} M\right)^{-\frac 1 2}$$

This may be combined with the Babylonian method for extracting the square root of a matrix to give a recurrence which converges to an orthogonal matrix quadratically: $$Q_{n + 1} = 2 M \left(Q_n^{-1} M + M^\mathrm{T} Q_n\right)^{-1}$$ where $Q_{0} = M$.

These iterations are stable provided the condition number of $M$ is less than three.

Using a first-order approximation of the inverse and the same initialization results in the modified iteration:

$$N_{n} = Q_n^\mathrm{T} Q_n$$ $$P_{n} = \frac 1 2 Q_n N_{n}$$ $$Q_{n + 1} = 2 Q_n + P_n N_n - 3 P_n$$

Spin and pin
A subtle technical problem afflicts some uses of orthogonal matrices. Not only are the group components with determinant +1 and −1 not connected to each other, even the +1 component, $SO(n)$, is not simply connected (except for SO(1), which is trivial). Thus it is sometimes advantageous, or even necessary, to work with a covering group of SO(n), the spin group, $Spin(n)$. Likewise, $O(n)$ has covering groups, the pin groups, Pin(n). For $n > 2$, $Spin(n)$ is simply connected and thus the universal covering group for $SO(n)$. By far the most famous example of a spin group is $Spin(3)$, which is nothing but $SU(2)$, or the group of unit quaternions.

The Pin and Spin groups are found within Clifford algebras, which themselves can be built from orthogonal matrices.

Rectangular matrices
If $Q$ is not a square matrix, then the conditions $Q^{T}Q = I$ and $QQ^{T} = I$ are not equivalent. The condition $Q^{T}Q = I$ says that the columns of Q are orthonormal. This can only happen if $Q$ is an $m × n$ matrix with $n ≤ m$ (due to linear dependence). Similarly, $QQ^{T} = I$ says that the rows of $Q$ are orthonormal, which requires $n ≥ m$.

There is no standard terminology for these matrices. They are variously called "semi-orthogonal matrices", "orthonormal matrices", "orthogonal matrices", and sometimes simply "matrices with orthonormal rows/columns".

For the case $n ≤ m$, matrices with orthonormal columns may be referred to as orthogonal k-frames and they are elements of the Stiefel manifold.

title: Orthogonal matrix


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Music of Spain

In Spain, music has a long history. It has played an important role in the development of Western music, and has greatly influenced Latin American music. Spanish music is often associated with traditional styles such as flamenco and classical guitar. While these forms of music are common, there are many different traditional musical and dance styles across the regions. For example, music from the north-west regions is heavily reliant on bagpipes, the jota is widespread in the centre and north of the country, and flamenco originated in the south. Spanish music played a notable part in the early developments of western classical music, from the 15th through the early 17th century. The breadth of musical innovation can be seen in composers like Tomás Luis de Victoria, styles like the zarzuela of Spanish opera, the ballet of Manuel de Falla, and the classical guitar music of Francisco Tárrega. Nowadays, in Spain as elsewhere, the different styles of commercial popular music are dominant.

Origins of the music of Spain

The Iberian peninsula has had a history of receiving different musical influences from around the Mediterranean Sea and across Europe. In the two centuries before the Christian era, Roman rule brought with it the music and ideas of Ancient Greece; early Christians, who had their own differing versions of church music arrived during the height of the Roman Empire; the Visigoths, a Romanized Germanic people, who took control of the peninsula following the fall of the Roman Empire; the Moors and Jews in the Middle Ages. Hence, there have been more than two thousand years of internal and external influences and developments that have produced a large number of unique musical traditions.

Medieval period
Isidore of Seville wrote about the local music in the 6th century. His influences were predominantly Greek, and yet he was an original thinker, and recorded some of the first details about the early music of the Christian church. He perhaps is most famous in musical history for declaring that it was not possible to notate sounds, an assertion which revealed his ignorance of the notational system of ancient Greece, suggesting that this knowledge had been lost with the fall of the Roman Empire in the west.

The Moors of Al-Andalus were usually relatively tolerant of Christianity and Judaism, especially during the first three centuries of their long presence in the Iberian peninsula, during which Christian and Jewish music continued to flourish. Music notation was developed in Spain as early as the 8th century (the so-called Visigothic neumes) to notate the chant and other sacred music of the Christian church, but this obscure notation has not yet been deciphered by scholars, and exists only in small fragments. The music of the early medieval Christian church in Spain is known, misleadingly, as the "Mozarabic Chant", which developed in isolation prior to the Islamic invasion and was not subject to the Papacy's enforcement of the Gregorian chant as the standard around the time of Charlemagne, by which time the Muslim armies had conquered most of the Iberian peninsula. As the Christian reconquista progressed, these chants were almost entirely replaced by the Gregorian standard, once Rome had regained control of the Iberian churches. The style of Spanish popular songs of the time is presumed to have been heavily influenced by the music of the Moors, especially in the south, but as much of the country still spoke various Latin dialects while under Moorish rule (known today as the Mozarabic) earlier musical folk styles from the pre-Islamic period continued in the countryside where most of the population lived, in the same way as the Mozarabic Chant continued to flourish in the churches. In the royal Christian courts of the reconquistors, music like the Cantigas de Santa Maria, also reflected Moorish influences. Other important medieval sources include the Codex Calixtinus collection from Santiago de Compostela and the Codex Las Huelgas from Burgos. The so-called Llibre Vermell de Montserrat (red book) is an important devotional collection from the 14th century.

Renaissance and Baroque periods
In the early Renaissance, Mateo Flecha el Viejo and the Castilian dramatist Juan del Encina ranked among the main composers in the post-Ars Nova period. Renaissance song books included the Cancionero de Palacio, the Cancionero de Medinaceli, the Cancionero de Upsala (kept in Carolina Rediviva library), the Cancionero de la Colombina, and the later Cancionero de la Sablonara. The organist Antonio de Cabezón stands out for his keyboard compositions and mastery.

An early 16th-century polyphonic vocal style developed in Spain was closely related to that of the Franco-Flemish composers. Merging of these styles occurred during the period when the Holy Roman Empire and the Burgundy were part of the dominions under Charles I (king of Spain from 1516 to 1556), since composers from the North of Europe visited Spain, and native Spaniards traveled within the empire, which extended to the Netherlands, Germany and Italy. Music composed for the vihuela by Luis de Milán, Alonso Mudarra and Luis de Narváez was one of the main achievements of the period. The Aragonese Gaspar Sanz authored the first learning method for guitar. Spanish composers of the Renaissance included Francisco Guerrero, Cristóbal de Morales, and Tomás Luis de Victoria (late Renaissance period), all of whom spent a significant portion of their careers in Rome. The latter was said to have reached a level of polyphonic perfection and expressive intensity equal or even superior to Palestrina and Lassus. Most Spanish composers returned home from travels abroad late in their careers to spread their musical knowledge in their native land, or in the late 16th century to serve at the Court of Philip II.

18th to 20th centuries

By the end of the 17th century the "classical" musical culture of Spain was in decline, and was to remain that way until the 19th century. Classicism in Spain, when it arrived, was inspired by Italian models, as in the works of Antonio Soler. Some outstanding Italian composers such as Domenico Scarlatti and Luigi Boccherini were appointed to the Madrid royal court. The short-lived Juan Crisóstomo Arriaga is credited as the main beginner of Romantic sinfonism in Spain.

Although symphonic music was never too important in Spain, chamber, solo instrumental (mainly guitar and piano) vocal and opera (both traditional opera, and the Spanish version of the singspiel) music was written by local composers. Zarzuela, a native form of opera that includes spoken dialogue, is a secular musical genre which developed in the mid-17th century, flourishing most importantly in the century after 1850. Francisco Asenjo Barbieri was a key figure in the development of the romantic zarzuela; whilst later composers such as Ruperto Chapí, Federico Chueca and Tomás Bretón brought the genre to its late 19th-century apogee. Leading 20th-century zarzuela composers included Pablo Sorozábal and Federico Moreno Torroba.

Fernando Sor, Dionisio Aguado, Francisco Tárrega and Miguel Llobet are known as composers of guitar music. Fine literature for violin was created by Pablo Sarasate and Jesús de Monasterio.

Musical creativity mainly moved into areas of popular music until the nationalist revival of the late Romantic era. Spanish composers of this period included Felipe Pedrell, Isaac Albéniz, Enrique Granados, Joaquín Turina, Manuel de Falla, Jesús Guridi, Ernesto Halffter, Federico Mompou, Salvador Bacarisse, and Joaquín Rodrigo.

In the 1930s, Spain's music scene was vibrant and diverse amidst significant social and political changes. Traditional Spanish music, particularly flamenco, continued to flourish, influenced by artists like Federico García Lorca and Manuel de Falla.

Performers

In the realm of classical music, Spain boasts a rich tradition of accomplished singers and performers who have made significant contributions to the global musical landscape. The country hosts a vibrant classical music scene, characterized by a multitude of professional orchestras and esteemed opera houses.

With over forty professional orchestras spread across its regions, Spain serves as a hub for orchestral excellence. Among these ensembles, notable names include the Orquestra Simfònica de Barcelona, Orquesta Nacional de España, and the Orquesta Sinfónica de Madrid. These orchestras showcase the nation's commitment to fostering musical talent and nurturing classical repertoire.

In addition to its orchestral prowess, Spain boasts a selection of distinguished opera houses that serve as pillars of the classical music community. Among these institutions, the Teatro Real, the Gran Teatre del Liceu, Teatro Arriaga, and the El Palau de les Arts Reina Sofía stand as esteemed venues renowned for their commitment to showcasing world-class opera productions. These opera houses provide platforms for both established and emerging singers to showcase their talents on a prestigious stage.

During the 1940s, Spanish music was shaped by the aftermath of the Civil War and Francisco Franco's dictatorship. Traditional genres like flamenco and classical music continued to thrive, albeit under strict censorship. Popular music forms such as zarzuela and pasodoble celebrated Spanish identity. The era reflected a complex interplay of cultural resilience, political control, and the influence of broader European events like World War II. Overall, music in 1940s Spain mirrored a society in transition, navigating between tradition and the constraints of authoritarian rule.

Ye-yé

The Yé-yé movement, stemming from the English pop-refrain "yeah-yeah," took on a unique form in various cultural contexts, notably in France and Spain. Initially coined in France to describe a genre influenced by American rock and British beat music of the early 1960s, such as the twist, Yé-yé music quickly gained popularity for its lively and uplifting characteristics. In Spain, Concha Velasco is often credited as the catalyst for the Yé-yé scene with her 1965 hit "La Chica Ye-Yé." However, even before Velasco's breakthrough, artists like Karina had already begun making significant strides with their infectious tunes as early as 1963. These pioneering female singers not only popularized Yé-yé music but also set the stage for a vibrant cultural movement characterized by catchy melodies and youthful exuberance.

Spanish Yé-yé music, much like its French counterpart, drew heavily from American and British influences but also incorporated elements of traditional Spanish music, particularly flamenco rhythms. This fusion of styles created a dynamic musical landscape that resonated with audiences not only in Spain but also across Spanish-speaking communities worldwide. The genre's ability to blend international pop trends with indigenous musical traditions contributed to its enduring appeal and cultural significance.

In contemporary times, artists such as Rosalía have continued to explore and reinterpret this fusion of styles, blending modern pop sensibilities with traditional Spanish music elements. This ongoing evolution highlights the Yé-yé movement's lasting influence on Spanish pop culture, showcasing its adaptability and continued relevance in shaping the musical identity of Spain. As a result, Yé-yé music remains a vital chapter in the broader narrative of how global musical influences intersect with local cultural expressions.

Popular music in Spain

Spanish pop music's journey through the tumultuous years of Francisco Franco's regime was marked by resilience and adaptation. Despite strict censorship and limited outlets for contemporary music, the Benidorm International Song Festival emerged as a beacon of opportunity for Spanish musicians. Inspired by Italy's San Remo Music Festival, Benidorm provided a crucial platform for artists to showcase new musical styles to eager audiences. This festival not only introduced international influences but also nurtured local talent, laying the groundwork for future developments in Spanish pop. An injured Real Madrid football player-turned-singer, for example, became the world-famous Julio Iglesias. During the 1960s and early 1970s, tourism boomed, bringing yet more musical styles from the rest of the continent and abroad. During Franco's rule, which heavily restricted women's rights and roles in public life, female artists faced additional barriers in expressing themselves through music. However, some managed to break through these constraints, contributing to the vibrant underground music scene that emerged despite censorship. Artists like Marisol, who started as a child star in the 1960s, and Rocío Dúrcal, who became a prominent figure in Spanish music and film, navigated these challenges to become beloved icons of their time.

The 1980s ushered in a transformative era with La Movida Madrileña, a vibrant cultural movement centered in Madrid. This period was pivotal for Spain's pop music scene, fostering a spirit of experimentation and creativity across diverse genres such as electronica, Euro disco, rock, punk, and hip-hop. Spanish pop music began to carve out its own distinctive identity, moving beyond emulation of Anglo-American trends to embrace originality and diversity. La Movida not only revitalized artistic expression but also catalyzed the industry's growth, setting the stage for Spain's emergence as a powerhouse in global music. Female artists such as Alaska (of Alaska y los Pegamoides and later Fangoria) and Martirio emerged as influential figures, challenging traditional gender roles and exploring new sounds and styles.

Julio Iglesias, Enrique Iglesias, and Alejandro Sanz stand as icons of Spanish pop music's international success. These artists not only captivated audiences within Spain but also resonated globally, showcasing the universal appeal of Spanish-language music. Julio Iglesias, in particular, achieved unprecedented success as the best-selling male Latin artist of all time, illustrating the enduring impact of Spanish pop on the global music landscape. In Spain itself, the 1990s were characterized by a vibrant underground music scene that thrived in major cities like Madrid and Barcelona. Alternative rock bands such as Los Planetas and Dover gained prominence, bringing a fresh energy to Spanish pop with their indie rock influences and experimental sounds. This period also saw the emergence of electronic music and dance clubs, further diversifying the musical landscape and fostering a new generation of DJs and producers. The integration of Spanish and Latin American music markets further amplified this influence, fostering a dynamic cultural exchange that continues to shape trends and innovation in the industry. Mónica Naranjo, known for her powerful vocals and dramatic performances, and Ana Belén, whose sophisticated style and versatile voice made her a prominent figure. These artists contributed to a diverse and vibrant era in Spanish music, each leaving a lasting impact with their distinctive contributions to the genre.

In contemporary times, Spanish female artists continue to thrive and innovate across various genres. Artists like Rosalía have garnered international acclaim for their unique blend of flamenco, pop, and urban music, pushing boundaries and redefining what it means to be a Spanish pop star in the global music scene. Aitana has risen to prominence through her participation in talent shows and subsequent chart-topping hits. These singers exemplify Spain's diverse musical landscape and its ability to blend traditional influences with contemporary trends in pop music. Male singers like Pablo Alborán, David Bisbal, Manuel Carrasco, and Melendi continue to captivate audiences with their distinct voices and memorable songs.

Artists like Enrique Iglesias and Alejandro Sanz have become successful internationally winning major music awards such as the Grammy Award. As Spanish is commonly spoken in Spain and most of Latin America, music from both regions have been able to crossover with each other. According to the Sociedad General de Autores y Editores (SGAE), Spain is the largest Latino music market in the world. As a result, the Latin music industry encompasses Spanish-language music from Spain. The Latin Academy of Recording Arts & Sciences, the organization responsible for the Latin Grammy Awards, includes music from Spain including a category for Best Flamenco Album with voting members living in the country.

Music by region
The regions of Spain have distinctive musical traditions. There is also a movement of singer-songwriters with politically active lyrics, paralleling similar developments in Latin America and Portugal. The singer and composer Eliseo Parra (b 1949) has recorded traditional folk music from the Basque country and Castile as well as his own compositions inspired from the musical styles of Spain and abroad.

Andalusia

Though Andalusia is best known for flamenco music, there is also a tradition of gaita rociera (tabor pipe) music in western Andalusia and a distinct violin and plucked-string type of band music known as panda de verdiales in Málaga.

Sevillanas is related to flamenco and most flamenco performers have at least one classic sevillana in their repertoire. The style originated as a medieval Castilian dance, called the seguidilla, which was adopted with a flamenco style in the 19th century. Today, this lively couples' dance is popular in most parts of Spain, though the dance is often associated with the city of Seville's famous Easter feria.

The region has also produced singer-songwriters like Javier Ruibal and Carlos Cano (singer-songwriter), who revived a traditional music called copla. Catalan Kiko Veneno and Joaquín Sabina are popular performers in a distinctly Spanish-style rock music, while Sephardic musicians like Aurora Moreno, Luís Delgado and Rosa Zaragoza keep Andalusian Sephardic music alive.

Aragon

Jota, popular across Spain, might have its historical roots in the southern part of Aragon. Jota instruments include the castanets, guitar, bandurria, tambourines and sometimes the flute. The guitarro, a unique kind of small guitar also seen in Murcia, seems Aragonese in origin. Besides its music for stick-dances and dulzaina (shawm), Aragon has its own gaita de boto (bagpipes) and chiflo (tabor pipe). As in the Basque country, Aragonese chiflo can be played along to a chicotén string-drum (psaltery) rhythm.

Asturias, Cantabria and Galicia

Northwest Spain (Asturias, Galicia and Cantabria) is home to a distinct musical tradition extending back into the Middle Ages. The signature instrument of the region is the gaita (bagpipe). The gaita is often accompanied by a snare drum, called the tamboril, and is played in processional marches. Other instruments include the requinta, a kind of fife, as well as harps, fiddles, rebec and zanfona (hurdy-gurdy). The music itself runs the gamut from uptempo muiñeiras to stately marches. As in the Basque Country, Cantabrian music also features intricate arch and stick dances but the tabor pipe does not play as an important role as it does in Basque music. Traditionally, Galician music included a type of chanting song known as alalas. Alalas may include instrumental interludes, and were believed to have a very long history, based on legends.

There are local festivals of which Ortigueira's Festival Internacional do Mundo Celta is especially important. Drum and bagpipe couples range among the most beloved kinds of Galician music, that also includes popular bands like Milladoiro. Pandereteiras are traditional groups of women that play tambourines and sing - bands like Tanxugueiras are directly influenced by this tradition. The bagpipe virtuosos Carlos Núñez and Susana Seivane are especially popular performers.

Asturias is also home to popular musicians such as José Ángel Hevia (bagpiper) and the group Llan de cubel. Circular dances using a 6/8 tambourine rhythm are a hallmark of this area. Vocal asturianadas show melismatic ornamentations similar to those of other parts of the Iberian Peninsula. There are many festivals, such as "Folixa na Primavera" (April, in Mieres), "Intercelticu d'Avilés" (Interceltic festival of Avilés, in July), as well as many "Celtic nights" in Asturias.

Balearic Islands
In the Balearic Islands, Xeremiers or colla de xeremiers are a traditional ensemble that consists of flabiol (a five-hole tabor pipe) and xeremias (bagpipes). Majorca's Maria del Mar Bonet was one of the most influential artists of nova canço, known for her political and social lyrics. Tomeu Penya, Biel Majoral, Cerebros Exprimidos and Joan Bibiloni are also popular.

Basque Country

The most popular kind of Basque music is named after the dance trikitixa, which is based on the accordion and tambourine. Popular performers are Joseba Tapia and Kepa Junkera. Highly appreciated folk instruments are the txistu (a tabor pipe similar to Occitanian galoubet recorder), alboka (a double clarinet played in circular-breathing technique, similar to other Mediterranean instruments like launeddas) and txalaparta (a huge xylophone, similar to the Romanian toacă and played by two performers in a fascinating game-performance). As in many parts of the Iberian peninsula, there are ritual dances with sticks, swords and arches made from vegetation. Other popular dances are the fandango, jota and 5/8 zortziko.

Basques on both sides of the Spanish-French border have been known for their singing since the Middle Ages, and a surge of Basque nationalism at the end of the 19th century led to the establishment of large Basque-language choirs that helped preserve their language and songs. Even during the persecution of the Francisco Franco era (1939–1975), when the Basque language was outlawed, traditional songs and dances were defiantly preserved in secret, and they continue to thrive despite the popularity of commercially marketed pop music.

Canary Islands
In the Canary Islands, Isa, a local kind of Jota, is now popular, and Latin American musical (Cuban) influences are quite widespread, especially with the charango (a kind of guitar). Timple, a local instrument which resembles ukulele / cavaquinho, is commonly seen in plucked-string bands. A popular set on El Hierro island consists of drums and wooden fifes (pito herreño). The tabor pipe is customary in some ritual dances on the island of Tenerife.

Castile, Madrid and León

A large inland region, Castile, Madrid and Leon were Celtiberian country before its annexation and cultural latinization by the Roman Empire but it is extremely doubtful that anything from the musical traditions of the Celtic era have survived. Ever since, the area has been a musical melting pot; including Roman, Visigothic, Jewish, Moorish, Italian, French and Roma influences, but the longstanding influences from the surrounding regions and Portugal continue to play an important role. Areas within Castile and León generally tend to have more musical affinity with neighboring regions than with more distant parts of the region. This has given the region diverse musical traditions.

Jota is popular, but is uniquely slow in Castile and León, unlike its more energetic Aragonese version. Instrumentation also varies much from the one in Aragon. Northern León, that shares a language relationship with a region in northern Portugal and the Spanish regions of Asturias and Galicia, also shares their musical influences. Here, the gaita (bagpipe) and tabor pipe playing traditions are prominent. In most of Castile, there is a strong tradition of dance music for dulzaina (shawm) and rondalla groups. Popular rhythms include 5/8 charrada and circle dances, jota and habas verdes. As in many other parts of the Iberian peninsula, ritual dances include paloteos (stick dances). Salamanca is known as the home of tuna, a serenade played with guitars and tambourines, mostly by students dressed in medieval clothing. Madrid is known for its chotis music, a local variation to the 19th-century schottische dance. Flamenco, although not considered native, is popular among some urbanites but is mainly confined to Madrid.

Catalonia
Though Catalonia is best known for sardana music played by a cobla, there are other traditional styles of dance music like ball de bastons (stick-dances), galops, ball de gitanes. Music is at the forefront in cercaviles and celebrations similar to Patum in Berga. Flabiol (a five-hole tabor pipe), gralla or dolçaina (a shawm) and sac de gemecs (a local bagpipe) are traditional folk instruments that make part of some coblas.

Catalan gipsies and Andalusian immigrants to Catalonia created their own style of rumba called rumba catalana which is a popular style that's similar to flamenco, but not technically part of the flamenco canon. The rumba catalana originated in Barcelona when the rumba and other Afro-Cuban styles arrived from Cuba in the 19th and early 20th centuries. Catalan performers adapted them to the flamenco format and made it their own. Though often dismissed by aficionados as "fake" flamenco, rumba catalana remains wildly popular to this day.

The havaneres singers remain popular. Nowadays, young people cultivate Rock català popular music, as some years ago the Nova Cançó was relevant.

Extremadura
Having long been the poorest part of Spain, Extremadura is a largely rural region known for the Portuguese influence on its music. As in the northern regions of Spain, there is a rich repertoire for tabor pipe music. The zambomba friction-drum (similar to Portuguese sarronca or Brazilian cuica) is played by pulling on a rope which is inside the drum. It is found throughout Spain. The jota is common, here played with triangles, castanets, guitars, tambourines, accordions and zambombas.

Murcia
Murcia is a region in the south-east of Spain which, historically, experienced considerable Moorish colonisation, is similar in many respects to its neighbour, Andalusia. The guitar-accompanied cante jondo Flamenco style is especially associated with Murcia as are rondallas, plucked-string bands. Christian songs, such as the polyphonic chant of the Auroro singers, are traditionally sung a cappella, sometimes accompanied by the sound of church bells, and cuadrillas are festive songs primarily played during holidays, like Christmas.

Navarre and La Rioja

Navarre and La Rioja are small northern regions with diverse cultural elements. Northern Navarre is Basque in language, while the Southern section shares more Aragonese features. The jota is also known in both Navarre and La Rioja. Both regions have rich dance and dulzaina (shawm) traditions. Txistu (tabor pipe) and dulzaina ensembles are very popular in the public celebrations of Navarre.

Valencia
Traditional music from Valencia is characteristically Mediterranean in origin. Valencia also has its local kind of Jota. Moreover, Valencia has a high reputation for musical innovation, and performing brass bands called bandes are common, with one appearing in almost every town. Dolçaina (shawm) is widely found. Valencia also shares some traditional dances with other Iberian areas, like for instance, the ball de bastons (stick-dances). The group Al Tall is also well-known, experimenting with the Berber band Muluk El Hwa, and revitalizing traditional Valencian music, following the Riproposta Italian musical movement.

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

title: Music of Spain


text:
Euler's criterion

In number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely,

Let p be an odd prime and a be an integer coprime to p. Then

a^{\tfrac{p-1}{2}} \equiv \begin{cases} \;\;\,1\pmod{p}& \text{ if there is an integer }x \text{ such that }x^2\equiv a \pmod{p},\\ -1\pmod{p}& \text{ if there is no such integer.} \end{cases} $$

Euler's criterion can be concisely reformulated using the Legendre symbol:

\left(\frac{a}{p}\right) \equiv a^{\tfrac{p-1}{2}} \pmod p. $$

The criterion dates from a 1748 paper by Leonhard Euler.

Proof
The proof uses the fact that the residue classes modulo a prime number are a field. See the article prime field for more details.

Because the modulus is prime, Lagrange's theorem applies: a polynomial of degree $k$ can only have at most $k$ roots. In particular, $x2 ≡ a (mod p)$ has at most 2 solutions for each $a$. This immediately implies that besides 0 there are at least $p − 1⁄2$ distinct quadratic residues modulo $p$: each of the $p − 1$ possible values of $x$ can only be accompanied by one other to give the same residue.

In fact, $$ (p-x)^{2}\equiv x^{2} \pmod p.$$This is because $$ (p-x)^{2} \equiv p^{2}-{2}{x}{p}+x^{2} \equiv x^{2} \pmod p.$$ So, the $$ \tfrac{p-1}{2}$$ distinct quadratic residues are: $$1^{2}, 2^{2}, ..., (\tfrac{p-1}{2})^{2} \pmod p. $$ As $a$ is coprime to $p$, Fermat's little theorem says that

a^{p-1}\equiv 1 \pmod p, $$ which can be written as

\left( a^{\tfrac{p-1}{2}}-1 \right)\left( a^{\tfrac{p-1}{2}}+1 \right)   \equiv 0 \pmod p. $$ Since the integers mod $p$ form a field, for each $a$, one or the other of these factors must be zero. Therefore,

a^{\tfrac{p-1}{2}}\equiv 1\pmod p $$ or

a^{\tfrac{p-1}{2}} \equiv {-1}\pmod p. $$

Now if $a$ is a quadratic residue, $a ≡ x^{2}$,

a^{\tfrac{p-1}{2}}\equiv {(x^2)}^{\tfrac{p-1}{2}} \equiv x^{p-1}\equiv1\pmod p. $$ So every quadratic residue (mod $p$) makes the first factor zero.

Applying Lagrange's theorem again, we note that there can be no more than $p − 1⁄2$ values of $a$ that make the first factor zero. But as we noted at the beginning, there are at least $p − 1⁄2$ distinct quadratic residues (mod $p$) (besides 0). Therefore, they are precisely the residue classes that make the first factor zero. The other $p − 1⁄2$ residue classes, the nonresidues, must make the second factor zero, or they would not satisfy Fermat's little theorem. This is Euler's criterion.

Alternative proof
This proof only uses the fact that any congruence $$kx\equiv l\!\!\! \pmod p$$ has a unique (modulo $$p$$) solution $$x$$ provided $$p$$ does not divide $$k$$. (This is true because as $$x$$ runs through all nonzero remainders modulo $$p$$ without repetitions, so does $$kx$$—if we have $$kx_1\equiv kx_2 \pmod p$$, then $$p\mid k(x_1-x_2)$$, hence $$p\mid (x_1-x_2)$$, but $$x_1$$ and $$x_2$$ aren't congruent modulo $$p$$.) It follows from this fact that all nonzero remainders modulo $$p$$ the square of which isn't congruent to $$a$$ can be grouped into unordered pairs $$(x,y)$$ according to the rule that the product of the members of each pair is congruent to $$a$$ modulo $$p$$ (since by this fact for every $$y$$ we can find such an $$x$$, uniquely, and vice versa, and they will differ from each other if $$y^2$$ is not congruent to $$a$$). If $$a$$ is not a quadratic nonresidue, this is simply a regrouping of all $$p-1$$ nonzero residues into $$(p-1)/2$$ pairs, hence we conclude that $$1\cdot2\cdot ... \cdot (p-1)\equiv a^{\frac{p-1}{2}} \!\!\! \pmod p$$. If $$a$$ is a quadratic residue, exactly two remainders were not among those paired, $$r$$ and $$-r$$ such that $$r^2\equiv a\!\!\! \pmod p$$. If we pair those two absent remainders together, their product will be $$-a$$ rather than $$a$$, whence in this case $$1\cdot2\cdot ... \cdot (p-1)\equiv -a^{\frac{p-1}{2}} \!\!\! \pmod p$$. In summary, considering these two cases we have demonstrated that for $$a\not\equiv 0 \!\!\! \pmod p$$ we have $$1\cdot2\cdot ... \cdot (p-1)\equiv -\left(\frac{a}{p}\right)a^{\frac{p-1}{2}} \!\!\! \pmod p$$. It remains to substitute $$a=1$$ (which is obviously a square) into this formula to obtain at once Wilson's theorem, Euler's criterion, and (by squaring both sides of Euler's criterion) Fermat's little theorem.

Examples
Example 1: Finding primes for which a is a residue

Let a = 17. For which primes p is 17 a quadratic residue?

We can test prime p's manually given the formula above.

In one case, testing p = 3, we have 17(3 − 1)/2 = 171 ≡ 2 ≡ −1 (mod 3), therefore 17 is not a quadratic residue modulo 3.

In another case, testing p = 13, we have 17(13 − 1)/2 = 176 ≡ 1 (mod 13), therefore 17 is a quadratic residue modulo 13. As confirmation, note that 17 ≡ 4 (mod 13), and 22 = 4.

We can do these calculations faster by using various modular arithmetic and Legendre symbol properties.

If we keep calculating the values, we find:
 * (17/p) = +1 for p = {13, 19, ...} (17 is a quadratic residue modulo these values)

 * (17/p) = −1 for p = {3, 5, 7, 11, 23, ...} (17 is not a quadratic residue modulo these values).

Example 2: Finding residues given a prime modulus p

Which numbers are squares modulo 17 (quadratic residues modulo 17)?

We can manually calculate it as:
 * 12 = 1
 * 22 = 4
 * 32 = 9
 * 42 = 16
 * 52 = 25 ≡ 8 (mod 17)
 * 62 = 36 ≡ 2 (mod 17)
 * 72 = 49 ≡ 15 (mod 17)
 * 82 = 64 ≡ 13 (mod 17).

So the set of the quadratic residues modulo 17 is {1,2,4,8,9,13,15,16}. Note that we did not need to calculate squares for the values 9 through 16, as they are all negatives of the previously squared values (e.g. 9 ≡ −8 (mod 17), so 92 ≡ (−8)2 = 64 ≡ 13 (mod 17)).

We can find quadratic residues or verify them using the above formula. To test if 2 is a quadratic residue modulo 17, we calculate 2(17 − 1)/2 = 28 ≡ 1 (mod 17), so it is a quadratic residue. To test if 3 is a quadratic residue modulo 17, we calculate 3(17 − 1)/2 = 38 ≡ 16 ≡ −1 (mod 17), so it is not a quadratic residue.

Euler's criterion is related to the law of quadratic reciprocity.

Applications
In practice, it is more efficient to use an extended variant of Euclid's algorithm to calculate the Jacobi symbol $$\left(\frac{a}{n}\right)$$. If $$n$$ is an odd prime, this is equal to the Legendre symbol, and decides whether $$a$$ is a quadratic residue modulo $$n$$.

On the other hand, since the equivalence of $$a^\frac{n-1}{2}$$ to the Jacobi symbol holds for all odd primes, but not necessarily for composite numbers, calculating both and comparing them can be used as a primality test, specifically the Solovay–Strassen primality test. Composite numbers for which the congruence holds for a given $$a$$ are called Euler–Jacobi pseudoprimes to base $$a$$.

title: Euler's criterion


text:
Baby Dodds

Warren "Baby" Dodds (December 24, 1898 – February 14, 1959) was an American jazz drummer born in New Orleans, Louisiana. He is regarded as one of the best jazz drummers of the pre-big band era. He varied his drum patterns with accents and flourishes, and he generally kept the beat with the bass drum while playing buzz rolls on the snare. Early influences included Louis Cottrell, Sr., Dave Perkins, and Tubby Hall. Dodds was among the first drummers to be recorded improvising while performing.

Early years
"Baby" Dodds was the younger brother of clarinetist Johnny Dodds. His mother, who died when he was nine years old, taught him valuable lessons about persistence and putting one's whole effort into endeavors, and he carried these with him through his career as a jazz drummer. He was born into a very musical family. His father and uncle played violin and his sister played harmonica. In addition, his father was religious and the family regularly sang hymns together. Dodds, in his autobiography The Baby Dodds Story, told the story of making his first drum: "I took a lard can and put holes in the bottom and turned it over and took nails and put holes around the top of it. Then I took some rungs out of my mother's chairs and made drumsticks out of them". At age 16, Dodds saved up enough money to buy his own drum set. Although Dodds had several paid teachers during his early years as a drummer, various jazz drummers around New Orleans also influenced him. He started playing in street parades around New Orleans with Bunk Johnson and his band and then gained a job playing in Willie Hightower's band, the American Stars. The band played in various venues around New Orleans, and Dodds recalled hearing many musicians along the way, including Buddy Bolden, John Robichaux, and Jelly Roll Morton. He played with several different outfits including those of Frankie Duson and Sonny Celestin, and he was part of the New Orleans tradition of playing jazz during funeral marches. Dodds describes this experience in his autobiography: "The jazz played after New Orleans funerals didn't show any lack of respect for the person being buried. It rather showed their people that we wanted them to be happy".

Prime years
Dodds gained a reputation as a top young drummer in New Orleans. In 1918, Dodds left Sonny Celestin's group to play in Fate Marable's riverboat band. A teenaged Louis Armstrong also joined the band, and the two of them were on the boats together. The band played on four different boats, and usually left New Orleans in May and traveled to St. Louis, though they also sometimes traveled further north. They played jazz, popular, and classical music while on the boats. Dodds and Armstrong left Fate Marable's band in 1921 due to a disagreement about musical style, and Dodds soon joined King Oliver's Creole Jazz Band. At this time, the personnel in Oliver's band were Joe "King" Oliver on cornet, Baby Dodds' brother Johnny Dodds on clarinet, Davey Jones on alto saxophone, Honoré Dutrey on trombone, Lil Hardin on piano, Jimmy Palao on violin, and Eddie Garland on bass fiddle. They moved to California in 1921 to work with Oliver there, and they played together for about fifteen months. In 1922, the band, excepting Garland, Palao, and Jones, followed Oliver to Chicago, which would be his base of operations for several years. They began playing at the Lincoln Gardens, and Armstrong also joined this outfit. Dodds describes playing with this band as "a beautiful experience". King Oliver's Creole Jazz Band broke up in 1924 due to disagreements about travel and musical style; the argument became so heated that the Dodds brothers threatened to beat up Oliver. Dodds recorded with Armstrong, Jelly Roll Morton, Art Hodes, and his brother Johnny Dodds. Dodds played in Armstrong's Hot Five and Hot Seven groups. In May 1927, Armstrong recorded with the Hot Seven, which consisted of Johnny Dodds, Johnny St. Cyr, Lil Hardin Armstrong, John Thomas, Pete Briggs, and Baby Dodds. From September to December 1927, the Hot Five Armstrong assembled consisted of Johnny Dodds, Kid Ory, Johnny St. Cyr, Lonnie Johnson, and Baby Dodds. During the decade, he also performed with the Black Bottom Stompers, Chicago Footwarmers, Willie Hightower, and Charlie Elgar.

Later years

During the 1930s, Dodds performed with Lil Hardin, Natty Dominique, and the Three Deuces's house band.

After the Oliver band broke up, the Dodds brothers played at Burt Kelley's Stables in Chicago, and soon after, Johnny Dodds began leading his own outfit, of which Baby was a part. Johnny Dodds died of a stroke in 1940. Of his brother, Baby Dodds said the following: "There just couldn't be another Johnny Dodds or anyone to take his place. And his passing on made a big difference in my life. I had been connected with him for many years and from then on I had to be wholly on my own". After his brother's death, Baby Dodds worked mostly as a freelance drummer around Chicago. This was the time of the New Orleans Revival, which was a movement in response to the emerging style of bebop. Many jazz traditionalists wanted jazz to return to its roots during this time. Dodds, having remained a New Orleans style drummer untouched by the influence of swing, found himself playing a role in the New Orleans jazz revival. In 1941, he played with Jimmie Noone and his band for a short time. This band featured Mada Roy on piano, Noone on clarinet, Bill Anderson on bass, and Dodds on drums. Dodds stayed with this outfit for only three months before they went to California, while Dodds decided to stay in Chicago. In the late 1940s he worked at Jimmy Ryan's in New York City. On some of his trips back to New Orleans, he recorded with Bunk Johnson. Dodds ended up playing with Johnson's band in New York. Dodds described his impressions of New York as a place where people listened to jazz rather than dancing to it: "When I first went to New York it seemed very strange to have people sitting around and listening rather than dancing. In a way, it was similar to theatre work. But it was peculiar for me because I always felt as though I was doing something for the people if they danced to the music". After playing with several outfits in New York, he joined Mezz Mezzrow's group on a tour of Europe in 1948 that lasted eight weeks. The group ended up playing solely in France, and Dodds had a great experience, saying that Europeans "take our kind of music much more seriously than they do in our own country". They played at the Nice Festival along with Rex Stewart, Louis Armstrong, and several other American jazz musicians.

Dodds returned to Chicago after the European tour and while taking a trip to New York in April 1949, he suffered a stroke. In 1950 he had his second stroke and in 1952 suffered a third. After his three strokes, Dodds tutored and played in public as much as he could, though he was unable to complete entire performances. He retired in 1957. He died on February 14, 1959, in Chicago, and was buried at Lincoln Cemetery in Blue Island, Illinois.

The relationship between Baby and Johnny
Several accounts of the Dodds brothers suggest that they did not always get along. When the brothers were younger, Johnny got a clarinet from his father but Baby did not get a drum even though he asked for one. In The Baby Dodds Story, Dodds discusses his jealousy of his older brother when they were children. As they grew up, Johnny refused to let Baby play music with him because Baby was a heavy drinker and Johnny did not drink. When Joe Oliver hired Baby and Johnny saw how much Baby's talent as a drummer had grown, however, Johnny changed his mind. Although they continued to argue about Baby's drinking habits, they grew closer as brothers and musicians, and as suggested above, Baby was greatly affected by his brother's death.

The validity of The Baby Dodds Story
In 1959, Larry Gara's The Baby Dodds Story was published for the first time. The revised edition was published in 1992. Several critics have questioned the validity of this book, given the fact that it is solely Baby Dodds giving an extensive interview, and therefore the events in the book are based on what Dodds himself remembers. In his introduction, Gara explains the interview process, which took place in 1953, every Sunday for about twelve weeks. Gara's wife transcribed the interview between Gara and Dodds, and Gara then edited the interviews. Gara wanted this book to be Baby Dodds telling his story rather than Gara telling it. Gara did, however, exhibit the same concerns that various critics have exhibited: one person may not remember things exactly as they happened. He, therefore, consulted jazz historian Bill Russell, who helped Gara check Dodds' interviews for details that historical findings did not support. The finished product is a narrative by Baby Dodds with minimal footnotes from Gara. Although some details may be exaggerated, The Baby Dodds Story serves as a depiction of early jazz and its many influences as seen through Dodds' eyes.

Recording sessions
In many of his recordings, Dodds had to use a wood block rather than a drum set because of the nature of the recording technology. It was therefore difficult to hear Dodds' original style from early recordings, such as his 1923 sessions with King Oliver's Creole Jazz Band. Dodds did, however, begin recording again in 1940, and by this time, the technology was able to show his talent on the drum set. Dodds is perhaps the first jazz drummer to record unaccompanied: in 1945 he recorded two solos for Circle Records, and the next year recorded a series of solos and reminiscences for Folkways Records. On his part of the record, Dodds discusses his drumming techniques and his drumset and playing examples of techniques. This record gives an idea of what Dodds would have sounded like in his prime had the recording technology been what it is today. In 1954, he played for a Natty Dominique recording session which also featured bassman Israel Crosby and pianist Lil Hardin Armstrong.

Dodds' drumming style

Dodds continues to be admired for the creativity of his playing. He believed in playing something different for every chorus of every tune. Most of his contemporaries played a short buzz or press roll on the backbeats (the 2nd and 4th beats), but Dodds played a long roll that lasted till the following beat, which created a smoother time feel that he later developed into the jazz ride pattern most commonly used ever since. Dodds was best known, however, for what he called his "shimmy beat", which he first used in 1918 at Jack Sheehan's in New Orleans. He described it in his autobiography: "One night a French soldier came in. When he heard the music he couldn't dance to it, but he just started to shake all over. That's the way it affected me. I saw him do it and I did it, too". Dodds' unique shimmy beat caught Louis Armstrong's eye as well, who said: "To watch him play, especially when he beat on the rim of his bass in a hot chorus, he sort of shimmied when he beat with his sticks. Oh! Boy that alone was in my estimation the whole worth of admission". Besides his unique drumming style, something important to Dodds was paying attention to the musicians in the outfit and fitting his drumming into the style the band was playing. He tried to get to know each member in the outfit and learn about how each person played his or her instrument. Throughout his autobiography, Dodds talks about listening to the different band members and using his role as a drummer to help the band come together: "It was my job to study each musician and give a different background for each instrument. When a man is playing it's up to the drummer to give him something to make him feel the music and make him work. That's the drummer's job".

Notable is the absence of the hi-hat cymbals and stands. Dodds never liked them: "I didn't like them and I still don't. Some drummers can't play without them. I can't play with them."

Baby Dodds also played washboard. He recorded together with his brother's washboard band occasionally (Victor BVE 46063-6).

Discography

 * "Jazz à La Creole" (Baby Dodds Trio)
 * "Baby Dodds"
 * "Talking and Drum Solos"
 * "Live At New York Town Hall 1947" (Mezz Mezzrow and Sidney Bechet-1947)
 * "Bunk Johnson- The King Of The Blue"
 * "Albert's Blues / Buddy Bolden Blues" Tell Record 29655 (78RPM)

Book

 * The Baby Dodds Story Edition: As Told to Larry Gara

Induction into the Downbeat Hall of Fame
In 2010, DownBeat magazine's Veterans Committee inducted Dodds into the DownBeat Hall of Fame. The Veterans Committee specifically looks at jazz artists who are no longer living who were overlooked for one reason or another while they were alive. The article in DownBeat about Dodds' induction again blames the recording technology in the 1920s, and it also acknowledges the fact that when Dodds was in his prime, the hi-hat had not yet been invented. By today's standards, Dodds played with an incomplete drum set for much of his career.

Further research
The Historic New Orleans Collection opened the Williams Research Center in 1966 as part of its plan to promote education about New Orleans history. The center has The Baby Dodds Collection on display, which features 334 items that document much of Dodds' career. Included in this collection are transcribed interviews from Larry Gara and Bill Russell as well as material for a movie about Dodds that Russell worked on.

title: Baby Dodds


text:
Gustavus, Alaska

Gustavus (Lingít: Wanachích T’aak Héen ) is a second-class city in Hoonah-Angoon Census Area in the U.S. state of Alaska. The 2020 census population was 655, which represents a 48% increase over the 2010 census of 442 and is one of the fastest growing communities in Alaska.

History
Gustavus, formerly known as "Strawberry Point", lies on the outwash plain created by the glaciers that once filled Glacier Bay. Two hundred years ago, it was primarily a single large "beach". The native Tlingit people and others used the area for fishing, berry picking, and other similar uses. The first settlers arrived in 1914, but left shortly afterward. The first permanent homestead was established in 1917, when Abraham Lincoln Parker moved his family to Strawberry Point. Many Gustavus residents are descendants and relatives of the original Parker homesteaders.

In 1925 the name became "Gustavus", when the U.S. Post Office required a change for its new post office, although locals continued calling it "Strawberry Point" long afterwards. The new name came from Point Gustavus at the mouth of Glacier Bay.

In 1793 George Vancouver named Point Adolphus (at the northern tip of Chichagof Island, and today a well-known humpback whale feeding area) after Adolphus Frederick, seventh son of King George III. In 1878, W. H. Dall, while working on a coastal survey, saw "Adolphus" on the map and assumed it was for Swedish king Gustavus Adolphus. The point across Icy Strait from Point Adolphus at the mouth of Glacier Bay was not named on the map, so Dall called it "Gustavus". Another possibility is that Dall named Gustavus for Gustavus C. Hanus, a U.S. Naval Academy graduate who had extensive experience throughout southeast Alaska, and both Dall and Hanus served with the Coast Survey in Alaska. Hanus laid out the first streets in Juneau and helped quell the trouble in Klukwan in 1881.

There is still a large beach at Gustavus, with many strawberries. The city is surrounded on three sides by Glacier Bay National Park and Preserve, and on the fourth side by water. The area is a temperate rain forest; spruce and hemlock trees reach heights of 60 m, and alders, balsam poplar, fern, mosses, fireweed, lupine, and other plants are also common. Gustavus's coastal location gives it a relatively mild winter. Summer temperatures range from 11 to 17 C; winter temperatures from -3 to 4 C.

Geography
Gustavus is located at 58.41639°N, -135.74556°W (58.416327, -135.745549).

According to the United States Census Bureau, the city has a total area of 95.7 km2, of which 85.0 sqkm are land and 10.7 sqkm, or 11.16%, are water.

Gustavus is split by the Salmon River, a small waterway crossed by a bridge for the paved road running out to Glacier Bay National Park. Coho salmon, Dolly Varden, and other fish are commonly caught in this river. Within the city limits, the river is affected by tides.

Climate
Gustavus has a wet, maritime subarctic climate (Köppen Dfc) with short, mild summers and long, cold winters.

Demographics
Gustavus first appeared on the 1940 U.S. Census as "Strawberry Point", an unincorporated village. In 1950, it returned as Gustavus. It was made a census-designated place (CDP) in 1980. Its status was changed to an incorporated city in 2004.

As of the census of 2000, there were 429 people, 199 households, and 114 families residing in the city. The population density was 11.4 PD/sqmi. There were 345 housing units at an average density of 9.2 /sqmi. The racial makeup of the city was 89% white, 0% Black or African American, 4% Native American, 0.2% Asian, 0.2% Pacific Islander, 2% from other races, and 4% from two or more races. 1.4% of the population were Hispanic or Latino of any race.

Of the 199 households, 28% had children under the age of 18 living with them, 47% were married couples living together, 6% had a female householder with no husband present, and 43% were non-families. 38% of all households were made up of individuals, and 6% had someone living alone who was 65 years of age or older. The average household size was 2.2 and the average family size was 2.9.

In the city, the population was spread out, with 26% under the age of 18, 3% from 18 to 24, 30% from 25 to 44, 36% from 45 to 64, and 5% who were 65 years of age or older. The median age was 40 years. For every 100 females, there were 130 males. For every 100 females age 18 and over, there were 135 males.

The median income for a household in the CDP was $34,800, and the median income for a family was $51,800. Males had a median income of $41,800 versus $29,400 for females. The per capita income for the city was $21,100. 15% of the population and 10% of families were below the poverty line. 13% of those under the age of 18 and 14% of those 65 and older were living below the poverty line.

Economy
The Gustavus economy is strongly linked to the surrounding natural resources; tourism and commercial fishing are mainstays. However, the importance of commercial fishing is in decline, as the Dungeness crab fishery in Glacier Bay National Park was closed in the early 2000s and the fishing for Dungeness outside of the national park has become poor. Salmon and halibut remain as the primary catch of the commercial fishing done out of Gustavus. Those not involved in tourism or seafood typically work for the government at the public school, the post office, the National Park Service, or the City of Gustavus.

The hardware and grocery store is called ToshCo, named after its owner and the fact that most of its inventory is from the Costco in Juneau.

Transportation
Gustavus is not connected to the contiguous highway system. Gustavus Airport and the Alaska Marine Highway provide access between Gustavus and the outside world. During the 2020 coronavirus pandemic, the ferry system shut down and Gustavus became even more isolated, relying on private boats for grocery shipments.

title: Gustavus, Alaska


text:
Hobart Bay, Alaska

Hobart Bay is a census-designated place that was located in Hoonah-Angoon Census Area, Alaska during the 2010 United States Census, but is now part of Petersburg Borough. As of the 2010 Census, the population of the CDP was one, down from three in 2000.

The bay was named in 1889 by Lieutenant Commander Mansfield of the United States Navy. The site was first settled as a logging camp. The population has declined in recent years. Nevertheless, on January 3, 2013, Hobart Bay and a sizable portion of the Hoonah-Angoon Census Area around it were incorporated into the new Petersburg Borough, which has the city of Petersburg as its economic center and is composed primarily of land from the former Petersburg Census Area.

Geography
Hobart Bay is located at 57°27'11" North, 133°23'36" West (57.453007, -133.393308).

According to the United States Census Bureau, the CDP has a total area of 129.8 sqmi, of which, 117.4 sqmi of it is land and 12.4 sqmi of it is water. The total area is 9.54% water.

Demographics
Hobart Bay first appeared on the 1990 U.S. Census as a census-designated place (CDP).

As of the census of 2000, there were three people, two households, and one family residing in the CDP. The population density was 0/km2. There were seventeen housing units at an average density of 0.1 /sqmi. The racial makeup of the CDP was two White and one Native American.

There were two households with no children under the age of eighteen living with them; one was a married couple living together, and one was a male individual.

In the CDP the population was spread out, with two from twenty-five to forty-four and one from forty-five to sixty-four. The median age was forty-four years. There was one female and two males.

The median income for a household in the CDP was $68,750, and the median income for a family was $68,750. The per capita income for the CDP was $34,900.

title: Hobart Bay, Alaska


text:
Hoonah, Alaska

Hoonah ( or Gaaw Yat’aḵ Aan) is a largely Tlingit community on Chichagof Island, located in Alaska's panhandle in the southeast region of the state. It is 30 mi west of Juneau, across the Alaskan Inside Passage. Hoonah is the only first-class city on Chichagof Island, the 109th-largest island in the world and the 5th-largest island in the U.S. At the 2020 census the population was 931, up from 760 in 2010. In the summer the population can swell to over 1,300 depending on fishing, boating, hiking and hunting conditions.

"Hoonah" became the official spelling in 1901, with establishment of the Hoonah branch of the United States Post Office. "Hoonah" is the approximate pronunciation of the Tlingit name Xunaa, which means “lee of the north wind”, i.e., protected from the north wind.

History
The clans comprising the Huna Tlingit originally settled what is now Glacier Bay National Park as well as Icy Strait, Cross Sound, and the outer coast north to Sea Otter Creek. Two catastrophic events forced the Tlingit from their homeland; rapid glacial advance in Glacier Bay and a landslide-induced tsunami in Lituya Bay along the outer coast. Tlingit oral tradition recounts these events as well as the clans' ultimate resettlement in Xunaa.

A partial timeline of modern Hoonah history is below:

 * 1750s - Xunaa was settled by clans fleeing glacial advance in Glacier Bay.
 * 1880 - The North West Trading Company built the first store in Hoonah.
 * 1881 - The Presbyterian Home Mission and school was built.
 * 1887 - 450 to 500 people were wintering in the village.
 * 1901 - Hoonah post office was opened.
 * 1912 - The Hoonah Packing Co. built a large cannery north of town. The cannery was shut down after decades of use and is now a tourist attraction at Icy Strait Point.
 * 1934, 1936 - Hoonah Indian Association was constituted as a federally recognized tribe, authorized to act on behalf of the Huna Tlingit.
 * 1944 - A fire destroyed much of Hoonah, including many priceless Tlingit cultural objects. The United States federal government assisted in rebuilding Hoonah.
 * 1953 - Last year that Icy Strait Salmon Company operated as a full-fledged canning operation. From this point until 1999 the property functioned as a maintenance and support facility for the seine boat fishing fleet.
 * 1964 - First graduating class from Hoonah High School, in May.
 * 2001 - Groundbreaking ceremony for Icy Strait Point, at abandoned Hoonah Packing Company site.
 * 2004 - Celebrity MV Mercury makes inaugural call at Icy Strait Point.
 * 2007 - ZipRider! The world's largest zipline opens at Icy Strait Point.
 * 2010 - Alaska State Troopers assisted Hoonah police in a 2-day manhunt for a man who was reported to be armed in a wooded area in the city. He was later apprehended without incident.
 * 2010 - A fire and subsequent series of explosions destroyed the Icy Strait Lumber Mill. No criminal activity was suspected in the fire.
 * 2010 - Two police officers from the city were shot and killed in an ambush on Front Street. The suspect subsequently barricaded himself inside his own home on August 29, 2010. State, local and federal law enforcement officers and SWAT personnel from agencies throughout Alaska responded in the wake of the shooting. The suspect, John Marvin, was taken into custody the next day.

The town of Hoonah is featured on the Discovery Channel show Alaskan Bush People.

Education and schools
Sheldon Jackson established the first school house and teacher's residence in November 1881. The school was initially overseen by Mr. and Mrs. Walter B. Styles of New York until their transfer to Sitka in 1882. The Reverend and Mrs. John McFarland assumed responsibility for the school in 1884, and by 1885 219 Tlingit students were enrolled: 69 boys, 76 girls and 74 adults. A territorial school and government school were built in 1923. In 1932 the government school was demolished and replaced by a Bureau of Indian Affairs school.

Hoonah City Schools currently serves the needs of Hoonah's elementary and secondary students. Six graduating seniors made up the Hoonah City Schools class of 2015.

Hoonah Packing Company
The Hoonah Packing Company ("HPC"), built in 1912, was one of eight canneries operating in the area during the early twentieth century, representing Hoonah's major industry at the time. HPC was sold several times until coming to be owned by Wards Cove Packing, which also owned Hoonah Trading. The cannery ceased operating in 1954, but continued to see use by commercial fishermen for storing and repairing their boats and gear.

Geography
Hoonah is on the north shore of Chichagof Island, on Icy Strait, at 58.10944°N, -135.43639°W (58.109435, -135.436349). The communities of Whitestone Logging Camp (which was being dismantled in early 2011) and Game Creek just south of the city limits. The port at Hoonah is called Port Frederick. Other incorporated communities nearby on Chichagof Island include Tenakee Springs to the south and Pelican to the west. A study began in 2009 regarding the feasibility of a road from Hoonah to Pelican and possibly connecting to Tenakee Springs to allow an energy corridor to hot spring thermal energy sources in the region for Hoonah, to lower heating and energy costs. Most Tenakee residents expressed opposition to the road, while Pelican has generally supported the idea. The road would save the Alaska Department of Transportation ferry costs in summer snow-free months, by connecting these areas to Hoonah.

According to the United States Census Bureau, the city of Hoonah has a total area of 18.9 km2, of which 15.6 km2 are land and 3.3 km2, or 17.59%, are water.

Tongass National Forest borders the area and has an unpaved road system of over 300 mi. Recreation areas include Game Creek, Kennel Creek, and Freshwater Bay which has a small boat harbor, all to the east; and Whitestone boat landing and False Bay recreation area to the southeast. These areas are inaccessible in winter due to deep snow.

Climate
According to the Köppen climate classification system, Hoonah has a humid continental climate (Dfb), or an oceanic climate (Cfb) depending on the isotherm used.

Demographics
Hoonah first reported on the 1880 U.S. Census as the Tlingit settlement of Koudekan. It reported as "Huna" on the 1890 U.S. Census. It next appeared on the 1900 U.S. Census as "Hooniah." From 1910 onwards, it appeared as Hoonah. It formally incorporated in 1946.

Hoonah is the principal village for the Huna Tlingit who originally settled Glacier Bay, Icy Strait, Cross Sound, and the Outer Coast. The four original Tlingit clans present are Chookaneidi, T'aakdeintaan, Wooshkeetaan, and Kaagwaantaan. Numerous other clans migrated to, or married into, the community, as have non-native peoples.

As of the census of 2000, there were 860 people, 300 households, and 215 families residing in the city. The population density was 130.2 PD/sqmi. There were 348 housing units at an average density of 52.7 /sqmi. The racial makeup of the city was 28.72% White, 0.23% Black or African American, 60.58% Native American, 0.12% Asian, 0.81% from other races, and 9.53% from two or more races. 3.60% of the population were Hispanic or Latino of any race.

There were 300 households, out of which 33.7% had children under the age of 18 living with them, 52.3% were married couples living together, 12.3% had a female householder with no husband present, and 28.3% were non-families. 22.7% of all households were made up of individuals, and 5.0% had someone living alone who was 65 years of age or older. The average household size was 2.83 and the average family size was 3.34.

In the city, the age distribution of the population shows 29.2% under the age of 18, 9.4% from 18 to 24, 26.5% from 25 to 44, 27.3% from 45 to 64, and 7.6% who were 65 years of age or older. The median age was 36 years. For every 100 females, there were 112.9 males. For every 100 females age 18 and over, there were 115.2 males.

The median income for a household in the city was $39,028, and the median income for a family was $45,125. Males had a median income of $37,083 versus $23,958 for females. The per capita income for the city was $16,097. About 14.3% of families and 16.6% of the population were below the poverty line, including 14.1% of those under age 18 and 10.4% of those age 65 or over.

The old fish cannery, which ceased operations in the 1950s, located near Icy Strait, was obtained by the Huna Totem Corporation (HTC), the village ANCSA Corporation. The road to the site, Cannery Road, was paved in 2000 and the site was converted into a tourism destination for cruise ship passengers. From May to September, cruise ships anchor off Icy Strait Point, and visits from ship passengers enhance Hoonah's warm-weather economy weekly. The former Hoonah Air Force Station, once a White Alice Communications System facility during the Cold War, which closed in the mid-1970s, is now the start point of a zip-line, one of the longest in the world, which ends at the cannery site. The cruise ship passengers, visiting fishing vessels, and summertime boaters who dock in the Hoonah city small boat harbor, all bring revenue to the city. The closing of the logging industry in southeast Alaska hurt the town economically in the early 1990s, but limited logging, tourism and fishing have helped to replace the void.

Hunters, hikers, campers, boaters and fishers all visit Hoonah as tourists throughout the year. The mild weather, much like that of Seattle, attracts tourists to the city.

Transportation

Hoonah, being an island community, is only accessible by boat or plane.

Ferry
The Alaska Marine Highway serves Hoonah with the M/V LeConte with recent gaps in coverage due to state funding and needed repair work. The ferry service has traditionally offered residents a slower but more dependable and cheaper option to travel to and from Hoonah to Juneau.

The city of Hoonah operates a small boat harbor, a large vessel mooring harbor and a new boat haul-out facility. The Alaska Department of Transportation built a new ferry facility that opened in early 2001 in Hoonah.

Airport
The Hoonah Airport was expanded in 2011 and now has a 3000 ft runway. The airport is planned for expansion to better allow military C-130s from the Coast Guard and Air National Guard to land in Hoonah.

The airport offers service via bush carrier Alaska Seaplanes which offers multiple flights a day between Hoonah and Juneau and other local communities. Connections can be made in Juneau with Alaska Airlines for regional and interstate travel or to other bush carriers traveling to remote villages or communities from Juneau.

Infrastructure
Hoonah has a K-12 school with approximately 120 students. There are two grocery stores, a fuel dealership, hardware store, bar, hotel and restaurant-bar, two cafes, an auto service center, several gift shops, several bed and breakfasts, a U.S. post office, a regional U.S. Forest Service office for Tongass National Forest, a fish processing facility, and a sporting goods store. In 2015, a local brewery was established under the name Hoonah Brewing Company. Plans are underway to build a new jail.

The city is an Alaskan first class city and provides all municipal services including police, utilities and road maintenance. The city also maintains a city park, built in 2010 near the harbor, and a youth activity center.

The police department has a five-bed jail and employs four paid police officers, along with volunteer reserve officers. The Hoonah volunteer Emergency Medical Service (EMS) was recognized by the state of Alaska in 2009 for excellence, and the Hoonah Volunteer Fire Department was accredited by the Alaska Fire Commission in 2010. The Alaska State Troopers have an office post in Hoonah, with one "brown shirt" wildlife enforcement trooper-pilot posted there.

The Alaska Courts maintain a court house in Hoonah for district court, presided over by a magistrate.

The Hoonah Indian Association maintains a tribal office, a senior citizens center and other services to local tribal members.

Health care
The SEARHC Hoonah Health Center (or SEARHC Hoonah Medical Center) is the primary health clinic in the community. Its senior staff are Physician Assistants and Nurse Practitioners. The facility operates as part of the Southeast Alaska Regional Health Consortium (or SEARHC). It is open Monday through Friday for primary care and acute concerns, and offers assistance after hours in emergency situations. In the summer of 2015, the health center was moved to a new, larger facility. Bartlett Regional Hospital in Juneau, roughly 40 miles away (or an approximately 20 minute flight), is the nearest hospital. Because no roads lead in or out of Hoonah, individuals in need of dire medical attention are often airlifted to Juneau or Sitka via air ambulance or Coast Guard helicopter. The community is also served by local Emergency Medical Service. The volunteer EMS team retains two fully functioning ambulances as well as a modified ambulance for off-road emergencies.

Religion
Hoonah has eight churches:

 * Sacred Heart Parish (Roman Catholic)
 * St. Nicholas Eastern Orthodox Church
 * Harbor of Faith Lutheran Church
 * Hoonah United Presbyterian Church (ECO)
 * Abundant Life Christian Fellowship (Assemblies of God)
 * Hoonah Pentecostal Church (United Pentecostal)
 * The Salvation Army Corps, Center for Worship and Service
 * Hoonah Baptist Fellowship

title: Hoonah, Alaska


text:
Klukwan, Alaska

Klukwan is a census-designated place (CDP) in Alaska, United States. It is technically in Hoonah-Angoon Census Area, though it is an enclave of Haines Borough. At the 2010 census the population was 95, down from 139 at the 2000 census.

History
Klukwan began as a Chilkat Tlingit village along the trade route later known as the Dalton Trail. In 1880, the U. S. Navy reported the name of the village as "Chilcat of Klukquan". The name is Tlakw Áan in Tlingit, meaning roughly "forever village" due to its antiquity. Klukwan is the only remaining of five Chilkat villages that were in the area before 1900.

Geography
Klukwan is located at 59.4°N, -135.89333°W (59.400098, -135.893393). It is 21 mi northwest of Haines, on the north side of the Chilkat River near the Haines Highway. It is bordered to the west by the Covenant Life CDP.

According to the United States Census Bureau, the Klukwan CDP has a total area of 4.9 km2, of which 3.6 km2 are land and 1.3 km2, or 26.55%, are water.

Klukwan is an enclave of Haines Borough, Alaska. Although Klukwan is located in the Hoonah-Angoon Census Area of the Unorganized Borough, it is entirely surrounded by territory of Haines Borough, and therefore cut off from the rest of the Census Area.

Demographics
Klukwan first appeared on the 1880 U.S. Census as "Kluckquan", an unincorporated Tlingit village. With 565 residents, it was recorded as the 5th largest community in Alaska. In 1890, it reported as "Klakwan." It next appeared in 1910 under its present spelling of Klukwan. It was made a census-designated place (CDP) as of the 1980 census.

As of the census of 2000, there were 139 people, 44 households, and 31 families residing in the CDP. The population density was 111.9 PD/sqmi. There were 85 housing units at an average density of 68.4 /sqmi. The racial makeup of the CDP was 10.07% White, 88.49% Native American, 0.72% Asian, and 0.72% from two or more races. 3.60% of the population were Hispanic or Latino of any race.

There were 44 households, out of which 54.5% had children under the age of 18 living with them, 45.5% were married couples living together, 15.9% had a female householder with no husband present, and 27.3% were non-families. 25.0% of all households were made up of individuals, and 6.8% had someone living alone who was 65 years of age or older. The average household size was 3.16 and the average family size was 3.75.

In the CDP, the age distribution of the population shows 33.1% under the age of 18, 13.7% from 18 to 24, 23.7% from 25 to 44, 23.0% from 45 to 64, and 6.5% who were 65 years of age or older. The median age was 27 years. For every 100 females, there were 104.4 males. For every 100 females age 18 and over, there were 121.4 males.

The median income for a household in the CDP was $30,714, and the median income for a family was $31,563. Males had a median income of $26,250 versus $26,250 for females. The per capita income for the CDP was $11,612. There were none of the families and 1.5% of the population living below the poverty line, including no under eighteens and none of those over 64.

title: Klukwan, Alaska