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:
- reflect A over its main diagonal (which runs from top-left to 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 i th row, j th column element of AT is the j th row, i th column element of A:
![[\mathbf{A}^\mathrm{T}]_{ij} = [\mathbf{A}]_{ji}](http://upload.wikimedia.org/math/6/7/6/676a09fb68a5cfb70409594b8622e226.png)
If A is an m × n matrix then AT is an n × m matrix.
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