# 2.2 Matrix Transposition

The transpose of an m × n matrix A is an n × m matrix denoted by AT. The columns of AT are the rows of A and the rows of AT are the columns of A.

 $a^{T}_{ij}=a_{ji},\operatorname{where}1\leq i\leq m\operatorname{and}1\leq j\leq n$ (6)

A symmetric matrix is its own transpose, i.e. if A is symmetric

 $\mathbf{A={A}^{T}}$

The transpose of the 2 × 3 matrix

 $\left(\begin{array}[]{ccc}{a}_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\end{array}\right)$

is the 3 × 2 matrix

 $\left(\begin{array}[]{cc}a_{11}&a_{21}\\ a_{12}&a_{22}\\ a_{13}&a_{23}\end{array}\right)$