In linear algebra, the transpose of a matrix is an operator that flips a matrix over its diagonal; that is, transposition switches the row and column indices of the matrix A to produce another matrix, often denoted A^T (among other notations).

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

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:

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

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

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

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

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