In linear algebra, the transpose of a matrix A is another matrix AT (also written A′, Atr, tA or At) created by any one of the following equivalent actions: \n* reflect A over its main diagonal (which runs from top-left to bottom-right) to obtain AT \n* write the rows of A as the columns of AT \n* write the columns of A as the rows of AT Formally, the i th row, j th column element of AT is the j th row, i th column element of A: If A is an m × n matrix then AT is an n × m matrix. The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley.

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