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.
| $$a^{T}_{ij}=a_{ji},\mbox{ where }1 \leq i \leq m \mbox{ and }1\leq j\leq n$$ | (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
| $$\left(\begin{array}[]{cc}a_{11}&a_{21}\\ a_{12}&a_{22}\\ a_{13}&a_{23}\end{array}\right)$$ |