Transpose

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

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 is an operator which flips a matrix over its diagonal, that is it switches the row and column indices of the matrix by producing another matrix denoted as A^T (also written A′, A^tr, ^tA or A^t). It is achieved 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:

$\left[\mathbf {A} ^{\operatorname {T} }\right]_{ij}=\left[\mathbf {A} \right]_{ji}.$

If A is an m × n matrix, then A^T is an n × m matrix. To avoid confusing the reader between the transpose operation and a matrix raised to the A^th power, the $\mathbf {A} ^{\top }$ symbol denotes the transpose operation.

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

Examples

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

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

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

Properties

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

1. $\left(\mathbf {A} ^{\operatorname {T} }\right)^{\operatorname {T} }=\mathbf {A} .$

   The operation of taking the transpose is an involution (self-inverse).

2. $\left(\mathbf {A} +\mathbf {B} \right)^{\operatorname {T} }=\mathbf {A} ^{\operatorname {T} }+\mathbf {B} ^{\operatorname {T} }.$
   The transpose respects addition.
3. $\left(\mathbf {AB} \right)^{\operatorname {T} }=\mathbf {B} ^{\operatorname {T} }\mathbf {A} ^{\operatorname {T} }.$
   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.
4. $\left(c\mathbf {A} \right)^{\operatorname {T} }=c\mathbf {A} ^{\operatorname {T} }.$
   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.
5. $\det \left(\mathbf {A} ^{\operatorname {T} }\right)=\det(\mathbf {A} ).$
   The determinant of a square matrix is the same as the determinant of its transpose.
6. The dot product of two column vectors a and b can be computed as the single entry of the matrix product:
   $\left[\mathbf {a} \cdot \mathbf {b} \right]=\mathbf {a} ^{\operatorname {T} }\mathbf {b} ,$
   which is written as a_i bⁱ in Einstein summation convention.
7. If A has only real entries, then A^TA is a positive-semidefinite matrix.
8. $\left(\mathbf {A} ^{\operatorname {T} }\right)^{-1}=\left(\mathbf {A} ^{-1}\right)^{\operatorname {T} }.$

   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.

9. If A is a square matrix, then its eigenvalues are equal to the eigenvalues of its transpose, since they share the same characteristic polynomial.

Special transpose matrices

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

$\mathbf {A} ^{\operatorname {T} }=\mathbf {A} .$

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

$\mathbf {A} ^{\operatorname {T} }=-\mathbf {A} .$

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

$\mathbf {A} ^{\operatorname {T} }={\overline {\mathbf {A} }}.$

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

$\mathbf {A} ^{\operatorname {T} }=-{\overline {\mathbf {A} }}.$

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

$\mathbf {A} ^{\operatorname {T} }=\mathbf {A} ^{-1}.$

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

$\mathbf {A} ^{\operatorname {T} }={\overline {\mathbf {A} ^{-1}}}.$

Products

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. Furthermore, these products are symmetric matrices. 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. But the columns of A^T are the rows of A, so the entry corresponds to the inner product of two rows of A. If p_{i j} is the entry of the product, it is obtained from rows i and j in A. The entry p_{j i} is also obtained from these rows, thus p_{i j} = p_{j i}, and the product matrix (p_{i j}) is symmetric. Similarly, the product A^T A is a symmetric matrix.

A quick proof of the symmetry of A A^T results from the fact that it is its own transpose:

$\left(\mathbf {A} \mathbf {A} ^{\operatorname {T} }\right)^{\operatorname {T} }=\left(\mathbf {A} ^{\operatorname {T} }\right)^{\operatorname {T} }\mathbf {A} ^{\operatorname {T} }=\mathbf {A} \mathbf {A} ^{\operatorname {T} }.$

Transpose of a linear map

See also: 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

${}^{\operatorname {T} }f:W^{*}\to V^{*}:\varphi \mapsto \varphi \circ f.$

Equivalently, the transpose ^tf is defined by the relation

$\left\langle \varphi ,f(v)\right\rangle =\left\langle {}^{\operatorname {T} }f(\varphi ),v\right\rangle \quad \forall \varphi \in W^{*},v\in V,$

where ⟨·,·⟩ is the natural pairing of each dual space with its respective vector space. This definition also applies unchanged to left modules and to vector spaces.

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(Ψ(w))(v), we find that B(v, w) = ^tB(w, v). Here, Ψ is the natural homomorphism V → V^∗∗ into the double dual.

Adjoint

Not to be confused with Hermitian adjoint.

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:

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

$B_{V}{\big (}v,g(w){\big )}=B_{W}{\big (}f(v),w{\big )}\quad \forall v\in V,w\in W.$

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

See also: In-place matrix transposition

Illustration of row- and column-major order

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.

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.

See also

• Conjugate transpose
• Moore–Penrose pseudoinverse
• Projection (linear algebra)

Further reading

• Maruskin, Jared M. (2012). Essential Linear Algebra. San José: Solar Crest. pp. 122–132. ISBN 978-0-9850627-3-6.
• Schwartz, Jacob T. (2001). Introduction to Matrices and Vectors. Mineola: Dover. pp. 126–132. ISBN 0-486-42000-0.

External links

• Gilbert Strang (Spring 2010) Linear Algebra from MIT Open Courseware
• Transpose, mathworld.wolfram.com