Transpose

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

Transpose of a matrix

Definition

The transpose of a matrix A, denoted by A^T, ^TA, A^tr, ^tA or A^t, may be constructed by any of the following methods:

1. Reflect A over its main diagonal (which runs from the top left to the bottom right) to obtain A^T
2. Write the rows of A as the columns of A^T
3. Write the columns of A as the rows of A^T

Formally, the ith row, jth column element of A^T is the jth row, ith column element of A:

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

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

Matrix definitions involving transposition

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

${\displaystyle {\mathbf{A}}^{\text{T}}=\mathbf{A}.}$

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

${\displaystyle {\mathbf{A}}^{\text{T}}=-\mathbf{A}.}$

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

${\displaystyle {\mathbf{A}}^{\text{T}}=\overline{\mathbf{A}}.}$

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

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

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

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

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

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

Examples

• ${\displaystyle {\left[\begin{array}{cc}1& 2\end{array}\right]}^{\text{T}}=\left[\begin{array}{c}1\\ 2\end{array}\right]}$
• ${\displaystyle {\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]}^{\text{T}}=\left[\begin{array}{cc}1& 3\\ 2& 4\end{array}\right]}$
• ${\displaystyle {\left[\begin{array}{cc}1& 2\\ 3& 4\\ 5& 6\end{array}\right]}^{\text{T}}=\left[\begin{array}{ccc}1& 3& 5\\ 2& 4& 6\end{array}\right]}$

Properties

Let A and B be matrices and c be a scalar.

• ${\displaystyle {\left({\mathbf{A}}^{\text{T}}\right)}^{\text{T}}=\mathbf{A}.}$

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

• ${\displaystyle {\left(\mathbf{A}+\mathbf{B}\right)}^{\text{T}}={\mathbf{A}}^{\text{T}}+{\mathbf{B}}^{\text{T}}.}$

  The transpose respects addition.

• ${\displaystyle {\left(c\mathbf{A}\right)}^{\text{T}}=c({\mathbf{A}}^{\text{T}}).}$

  The transpose of a scalar is the same scalar. Together with the preceding property, this implies that the transpose is a linear map from the space of m × n matrices to the space of the n × m matrices.

• ${\displaystyle {\left(\mathbf{A}\mathbf{B}\right)}^{\text{T}}={\mathbf{B}}^{\text{T}}{\mathbf{A}}^{\text{T}}.}$

  The order of the factors reverses. By induction, this result extends to the general case of multiple matrices, so

  (A₁A₂...A_k−1A_k)^T = A_k^TA_k−1^T…A₂^TA₁^T.

• ${\displaystyle det\left({\mathbf{A}}^{\text{T}}\right)=det(\mathbf{A}).}$

  The determinant of a square matrix is the same as the determinant of its transpose.

• The dot product of two column vectors a and b can be computed as the single entry of the matrix product$${\displaystyle \mathbf{a}\cdot \mathbf{b}={\mathbf{a}}^{\text{T}}\mathbf{b}.}$$
• If A has only real entries, then A^TA is a positive-semidefinite matrix.
• ${\displaystyle {\left({\mathbf{A}}^{\text{T}}\right)}^{-1}={\left({\mathbf{A}}^{-1}\right)}^{\text{T}}.}$

  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. This can also be seen directly, see Eigenvalues of Transpose
• ${\displaystyle \left(\mathbf{A}\mathbf{a}\right)\cdot \mathbf{b}=\mathbf{a}\cdot \left({\mathbf{A}}^{\text{T}}\mathbf{b}\right)}$ for two column vectors ${\displaystyle \mathbf{a},\mathbf{b}}$ and the standard dot product.
• Over any field ${\displaystyle k}$, a square matrix ${\displaystyle \mathbf{A}}$ is similar to ${\displaystyle {\mathbf{A}}^{\text{T}}}$.

  This implies that ${\displaystyle \mathbf{A}}$ and ${\displaystyle {\mathbf{A}}^{\text{T}}}$ have the same invariant factors, which implies they share the same minimal polynomial, characteristic polynomial, and eigenvalues, among other properties.

  A proof of this property uses the following two observations.

  • Let ${\displaystyle \mathbf{A}}$ and ${\displaystyle \mathbf{B}}$ be ${\displaystyle n\times n}$ matrices over some base field ${\displaystyle k}$ and let ${\displaystyle L}$ be a field extension of ${\displaystyle k}$. If ${\displaystyle \mathbf{A}}$ and ${\displaystyle \mathbf{B}}$ are similar as matrices over ${\displaystyle L}$, then they are similar over ${\displaystyle k}$. In particular this applies when ${\displaystyle L}$ is the algebraic closure of ${\displaystyle k}$.
  • If ${\displaystyle \mathbf{A}}$ is a matrix over an algebraically closed field in Jordan normal form with respect to some basis, then ${\displaystyle \mathbf{A}}$ is similar to ${\displaystyle {\mathbf{A}}^{\text{T}}}$. This further reduces to proving the same fact when ${\displaystyle \mathbf{A}}$ is a single Jordan block, which is a straightforward exercise.

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_ij is the entry of the product, it is obtained from rows i and j in A. The entry p_ji is also obtained from these rows, thus p_ij = p_ji, and the product matrix (p_ij) 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:

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

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.

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.