Matrix - Transpose

Matrix transposition is the operation of swapping the rows and columns of a matrix.

The transpose of a matrix $A$ is denoted by $A^T$.

If $A$ is an $m$ by $n$ matrix, then its transpose $A^T$ is an $n$ by $m$ matrix defined by

$$(A^T{)}_{i,j}=A_{j,i}\text{for }1\le i\le n,1\le j\le m$$

Properties

Matrix Multiplication

If $A$ is an $m$ by $n$ matrix and $B$ is an $n$ by $p$ matrix, then:

$$A^T\cdot B^T=(B\cdot A{)}^T$$