Transpose

matrix transposetranspositionmatrix transpositiontransposed matrixTransposingtransposedtranspositionsarray transpositiondual spacetranspose matrix
Note that this article assumes that matrices are taken over a commutative ring.wikipedia
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Skew-symmetric matrix

skew-symmetricskew-symmetric matricesantisymmetric matrix
A square matrix whose transpose is equal to its negative is called a skew-symmetric matrix; that is, A is skew-symmetric if
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric ) matrix is a square matrix whose transpose equals its negative.

Matrix (mathematics)

matrixmatricesmatrix theory
Equivalently, a matrix A is orthogonal if its transpose is equal to its inverse:

Conjugate transpose

Hermitian transposeHermitian transpositionadjoint matrix
A square complex matrix whose transpose is equal to the matrix with every entry replaced by its complex conjugate (denoted here with an overline) is called a Hermitian matrix (equivalent to the matrix being equal to its conjugate transpose); that is, A is Hermitian if
In mathematics, the conjugate transpose or Hermitian transpose of an m-by-n matrix with complex entries is the n-by-m matrix obtained from by taking the transpose and then taking the complex conjugate of each entry.

Orthogonal matrix

orthogonal matricesorthogonalorthogonal transform
A square matrix whose transpose is equal to its inverse is called an orthogonal matrix; that is, A is orthogonal if
This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse:

Symmetric matrix

symmetricsymmetric matricessymmetrical
A square matrix whose transpose is equal to itself is called a symmetric matrix; that is, A is symmetric if
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose.

Linear algebra

linearlinear algebraiclinear-algebraic
For example, software libraries for linear algebra, such as BLAS, typically provide options to specify that certain matrices are to be interpreted in transposed order to avoid the necessity of data movement.
If V and W are finite dimensional, and M is the matrix of f in terms of some ordered bases, then the matrix of f^* over the dual bases is the transpose M^\mathsf T of M, obtained by exchanging rows and columns.

Invertible matrix

invertibleinversenonsingular
A square matrix whose transpose is equal to its inverse is called an orthogonal matrix; that is, A is orthogonal if
: where |A| is the determinant of A, C is the matrix of cofactors, and C T represents the matrix transpose.

Complex number

complexreal partimaginary part
A square complex matrix whose transpose is equal to the matrix with every entry replaced by its complex conjugate (denoted here with an overline) is called a Hermitian matrix (equivalent to the matrix being equal to its conjugate transpose); that is, A is Hermitian if A square complex matrix whose transpose is equal to the negation of its complex conjugate is called a skew-Hermitian matrix; that is, A is skew-Hermitian if
:The conjugate \overline z corresponds to the transpose of the matrix.

Inner product space

inner productinner-product spaceinner products
Indeed, the matrix product A A T has entries that are the inner product of a row of A with a column of A T.
is the transpose of

Matrix multiplication

matrix productmultiplicationproduct
If A is an m × n matrix and A T is its transpose, then the result of matrix multiplication with these two matrices gives two square matrices: A A T is m × m and A T A is n × n.
:where is the row vector obtained by transposing \mathbf x and the resulting 1×1 matrix is identified with its unique entry.

Orthogonal group

special orthogonal grouprotation grouporthogonal
The adjoint allows us to consider whether g : W → V is equal to f −1 : W → V. In particular, this allows the orthogonal group over a vector space V with a quadratic form to be defined without reference to matrices (nor the components thereof) as the set of all linear maps V → V for which the adjoint equals the inverse.
orthogonal matrices, where the group operation is given by matrix multiplication; an orthogonal matrix is a real matrix whose inverse equals its transpose.

Dual space

dualdual vector spacecontinuous dual
into the double dual.
If f : V → W is a linear map, then the transpose (or dual) f : W → V is defined by

Bilinear form

bilinearperfect pairingskew-symmetric
If the vector spaces V and W have respectively nondegenerate bilinear forms B V and B W, a concept known as the adjoint, which is closely related to the transpose, may be defined:
One can then show that B 2 is the transpose of the linear map B 1 (if V is infinite-dimensional then B 2 is the transpose of B 1 restricted to the image of V in V ∗∗ ).

In-place matrix transposition

in-place algorithmsrearrangement in memoryrows and columns can be switched
Therefore, efficient in-place matrix transposition has been the subject of numerous research publications in computer science, starting in the late 1950s, and several algorithms have been developed.
In-place matrix transposition, also called in-situ matrix transposition, is the problem of transposing an N×M matrix in-place in computer memory, ideally with O(1) (bounded) additional storage, or at most with additional storage much less than NM.

Basic Linear Algebra Subprograms

BLASDGEMMGeneral Matrix Multiply
For example, software libraries for linear algebra, such as BLAS, typically provide options to specify that certain matrices are to be interpreted in transposed order to avoid the necessity of data movement.
can optionally be transposed or hermitian-conjugated inside the routine and all three matrices may be strided.

Isomorphism

isomorphicisomorphouscanonical isomorphism
These bilinear forms define an isomorphism between V and V ∗, and between W and W ∗, resulting in an isomorphism between the transpose and adjoint of f.
This corresponds to transforming a column vector (element of V) to a row vector (element of V*) by transpose, but a different choice of basis gives a different isomorphism: the isomorphism "depends on the choice of basis".

Fast Fourier transform

FFTFast Fourier Transform (FFT)Fast Fourier Transforms
If repeated operations need to be performed on the columns, for example in a fast Fourier transform algorithm, transposing the matrix in memory (to make the columns contiguous) may improve performance by increasing memory locality.
Yet another variation is to perform matrix transpositions in between transforming subsequent dimensions, so that the transforms operate on contiguous data; this is especially important for out-of-core and distributed memory situations where accessing non-contiguous data is extremely time-consuming.

Row- and column-major order

row-major ordercolumn-major orderrow-major
For example, with a matrix stored in row-major order, the rows of the matrix are contiguous in memory and the columns are discontiguous.
As exchanging the indices of an array is the essence of array transposition, an array stored as row-major but read as column-major (or vice versa) will appear transposed.

Hermitian adjoint

adjointadjoint operatorHermitian conjugate
The Hermitian adjoint of a map between such spaces is defined similarly, and the matrix of the Hermitian adjoint is given by the conjugate transpose matrix if the bases are orthonormal.

Arthur Cayley

CayleyA. CayleyCayley, Arthur
The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley.

Complex conjugate

complex conjugationconjugateconjugation
A square complex matrix whose transpose is equal to the matrix with every entry replaced by its complex conjugate (denoted here with an overline) is called a Hermitian matrix (equivalent to the matrix being equal to its conjugate transpose); that is, A is Hermitian if

Hermitian matrix

HermitianHermitian matricesHermitian conjugate matrix
A square complex matrix whose transpose is equal to the matrix with every entry replaced by its complex conjugate (denoted here with an overline) is called a Hermitian matrix (equivalent to the matrix being equal to its conjugate transpose); that is, A is Hermitian if

Skew-Hermitian matrix

skew-Hermitiananti-Hermitianskew-Hermitian matrices
A square complex matrix whose transpose is equal to the negation of its complex conjugate is called a skew-Hermitian matrix; that is, A is skew-Hermitian if

Unitary matrix

unitaryunitary matricesunitary operator
A square complex matrix whose transpose is equal to its conjugate inverse is called a unitary matrix; that is, A is unitary if