Transpose

This article is about the transpose of a matrix. For other uses, see Transposition

Note that this article assumes that matrices are taken over a commutative ring. These results may not hold in the non-commutative case.

The transpose A^T of a matrix A can be obtained by reflecting the elements along its main diagonal. Repeating the process on the transposed matrix returns the elements to their original position.

In linear algebra, the transpose of a matrix A is another matrix A^T (also written A′, A^tr, ^tA or A^t) created by any one of the following equivalent actions:

reflect A over its main diagonal (which runs from top-left to bottom-right) to obtain A^T
write the rows of A as the columns of A^T
write the columns of A as the rows of A^T

Formally, the i th row, j th column element of A^T is the j th row, i th column element of A:

[\mathbf{A}^\mathrm{T}]_{ij} = [\mathbf{A}]_{ji}

If A is an m × n matrix then A^T is an n × m matrix.

The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley.^[1]

Examples

$\begin{bmatrix} 1 & 2 \end{bmatrix}^{\mathrm{T}} = \, \begin{bmatrix} 1 \\ 2 \end{bmatrix}$

$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}^{\mathrm{T}} = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}$

$\begin{bmatrix} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{bmatrix}^{\mathrm{T}} = \begin{bmatrix} 1 & 3 & 5\\ 2 & 4 & 6 \end{bmatrix}$

Properties

For matrices A, B and scalar c we have the following properties of transpose:

$( \mathbf{A}^\mathrm{T} ) ^\mathrm{T} = \mathbf{A} \quad \,$
The operation of taking the transpose is an involution (self-inverse).
$(\mathbf{A}+\mathbf{B}) ^\mathrm{T} = \mathbf{A}^\mathrm{T} + \mathbf{B}^\mathrm{T} \,$
The transpose respects addition.
$\left( \mathbf{A B} \right) ^\mathrm{T} = \mathbf{B}^\mathrm{T} \mathbf{A}^\mathrm{T} \,$
Note that the order of the factors reverses. From this one can deduce that a square matrix A is invertible if and only if A^T is invertible, and in this case we have (A⁻¹)^T = (A^T)⁻¹. By induction this result extends to the general case of multiple matrices, where we find that (A₁A₂...A_k−1A_k)^T = A_k^TA_k−1^T...A₂^TA₁^T.
$(c \mathbf{A})^\mathrm{T} = c \mathbf{A}^\mathrm{T} \,$
The transpose of a scalar is the same scalar. Together with (2), this states that the transpose is a linear map from the space of m × n matrices to the space of all n × m matrices.
$\det(\mathbf{A}^\mathrm{T}) = \det(\mathbf{A}) \,$
The determinant of a square matrix is the same as that of its transpose.
The dot product of two column vectors a and b can be computed as the single entry of the matrix product
$\left[ \mathbf{a} \cdot \mathbf{b} \right] = \mathbf{a}^{\mathrm{T}} \mathbf{b} ,$
which is written as a_i bⁱ in Einstein summation convention.
If A has only real entries, then A^TA is a positive-semidefinite matrix.
$(\mathbf{A}^\mathrm{T})^{-1} = (\mathbf{A}^{-1})^\mathrm{T} \,$
The transpose of an invertible matrix is also invertible, and its inverse is the transpose of the inverse of the original matrix. The notation A^−T is sometimes used to represent either of these equivalent expressions.
If A is a square matrix, then its eigenvalues are equal to the eigenvalues of its transpose since they share the same characteristic polynomial.
$\mathbf{A} \mathbf{A}^\mathrm{T}$ is a symmetric matrix.^[2]

Special transpose matrices

A square matrix whose transpose is equal to itself is called a symmetric matrix; that is, A is symmetric if

\mathbf{A}^{\mathrm{T}} = \mathbf{A} .

A square matrix whose transpose is equal to its negative is called a skew-symmetric matrix; that is, A is skew-symmetric if

\mathbf{A}^{\mathrm{T}} = -\mathbf{A} .

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

\mathbf{A}^{\mathrm{T}} = \overline{\mathbf{A}} .

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

\mathbf{A}^{\mathrm{T}} = -\overline{\mathbf{A}} .

A square matrix whose transpose is equal to its inverse is called an orthogonal matrix; that is, A is orthogonal if

\mathbf{A}^{\mathrm{T}} = \mathbf{A}^{-1} .

Transpose of a linear map

The transpose may be defined more generally:

If f : V → W is a linear map between right R-modules V and W with respective dual modules V^∗ and W^∗, the transpose of f is the linear map

{}^\mathrm{t} f : W^* \to V^* : \varphi \mapsto \varphi \circ f .

Equivalently, the transpose ^tf is defined by the relation

\left\langle \phi , f ( v ) \right\rangle = \left\langle {}^\mathrm{t} f ( \phi ) , v \right\rangle \quad \forall \phi \in W^* , v \in V ,

where ⟨·,·⟩ is the duality pairing. This definition also applies unchanged to left modules and to vector spaces.^[3]

The definition of the transpose may be seen to be independent of any bilinear form on the vector spaces, unlike the adjoint (below).

If the matrix A describes a linear map with respect to bases of V and W, then the matrix A^T describes the transpose of that linear map with respect to the dual bases.

Transpose of a bilinear form

Main article: Bilinear form

Every linear map to the dual space f : V → V^∗ defines a bilinear form B : V × V → F, with the relation B(v, w) = f(v)(w). By defining the transpose of this bilinear form as the bilinear form ^tB defined by the transpose ^tf : V^∗∗ → V^∗ i.e. ^tB(w, v) = ^tf(w)(v), we find that B(v, w) = ^tB(w, v).

Adjoint

Not to be confused with Hermitian adjoint.

If the vector spaces V and W have respective nondegenerate bilinear forms B_V and B_W, a concept closely related to the transpose – the adjoint – may be defined:

If f : V → W is a linear map between vector spaces V and W, we define g as the adjoint of f if g : W → V satisfies

B_V(v, g(w)) = B_W(f(v),w) \quad \forall v \in V, w \in W .

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. The matrix of the adjoint of a map is the transposed matrix only if the bases are orthonormal with respect to their bilinear forms. In this context, many authors use the term transpose to refer to the adjoint as defined here.

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.

Over a complex vector space, one often works with sesquilinear forms (conjugate-linear in one argument) instead of bilinear forms. 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.

Implementation of matrix transposition on computers

On a computer, one can often avoid explicitly transposing a matrix in memory by simply accessing the same data in a different order. 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.

However, there remain a number of circumstances in which it is necessary or desirable to physically reorder a matrix in memory to its transposed ordering. For example, with a matrix stored in row-major order, the rows of the matrix are contiguous in memory and the columns are discontiguous. 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.

Main article: In-place matrix transposition

Ideally, one might hope to transpose a matrix with minimal additional storage. This leads to the problem of transposing an n × m matrix in-place, with O(1) additional storage or at most storage much less than mn. For n ≠ m, this involves a complicated permutation of the data elements that is non-trivial to implement in-place. 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.

References

↑ Arthur Cayley (1858) "A memoir on the theory of matrices," Philosophical Transactions of the Royal Society of London, 148 : 17-37. The transpose (or "transposition") is defined on page 31.
↑ "Is a matrix multiplied with its transpose something special?" Proof in answer on StackExchange.com,
↑ Bourbaki, "II §2.5", Algebra I

External links

MIT Linear Algebra Lecture on Matrix Transposes
Transpose, mathworld.wolfram.com
Transpose, planetmath.org
Khan Academy introduction to matrix transposes

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