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.

aijT=aji,where1imand1jna^{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

𝐀=𝐀𝐓\mathbf{A={A}^{T}}

The transpose of the 2 × 3 matrix

(a11a12a13a21a22a23)\left(\begin{array}[]{ccc}{a}_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\end{array}\right)

is the 3 × 2 matrix

(a11a21a12a22a13a23)\left(\begin{array}[]{cc}a_{11}&a_{21}\\ a_{12}&a_{22}\\ a_{13}&a_{23}\end{array}\right)