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.

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

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

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