In linear algebra, transposition is an operation that flips a matrix over its diagonal; that is, transposition switches the row and column indices of the matrix A to produce another matrix, called the transpose of A and often denoted AT (among other notations). 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 AT, TA, Atr, tA or At, may be constructed by any of the following methods:
Reflect A over its main diagonal (which runs from the top left to the bottom right) to obtain AT Write the rows of A as the columns of AT Write the columns of A as the rows of AT Formally, the ith row, jth column element of AT is the jth row, ith column element of A:
[ A T ] i j = [ A ] j i . {\displaystyle \left[\mathbf {A} ^{\text{T}}\right]_{ij}=\left[\mathbf {A} \right]_{ji}.}
If A is an m × n matrix, then AT 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
A T = A . {\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
A T = − A . {\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
A T = A ¯ . {\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
A T = − A ¯ . {\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
A T = A − 1 . {\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
A T = A − 1 ¯ . {\displaystyle \mathbf {A} ^{\text{T}}={\overline {\mathbf {A} ^{-1}}}.}
Examples
[ 1 2 ] T = [ 1 2 ] {\displaystyle {\begin{bmatrix}1&2\end{bmatrix}}^{\text{T}}=\,{\begin{bmatrix}1\\2\end{bmatrix}}}
[ 1 2 3 4 ] T = [ 1 3 2 4 ] {\displaystyle {\begin{bmatrix}1&2\\3&4\end{bmatrix}}^{\text{T}}={\begin{bmatrix}1&3\\2&4\end{bmatrix}}}
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