Matrix Transpose

linear-algebra

Definition

Matrix Transpose

The transpose of a matrix $M$ is the matrix with the rows and columns interchanged:

$$\begin{array}{rl}{M}^T& ={\left[\begin{array}{cccc}{a}_{11}& a_{12}& \dots & a_{1n}\\ a_{21}& a_{22}& \dots & a_{2n}\\ ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& \dots & a_{mn}\end{array}\right]}^T\\ & =\left[\begin{array}{cccc}{a}_{11}& a_{21}& \dots & a_{m1}\\ a_{12}& a_{22}& \dots & a_{m2}\\ ⋮& ⋮& \ddots & ⋮\\ a_{1n}& a_{2n}& \dots & a_{mn}\end{array}\right]\end{array}$$

Example

$$M^T={\left[\begin{array}{cc}1& 2\\ 5& 7\\ 0& 2\end{array}\right]}^T=\left[\begin{array}{ccc}1& 5& 0\\ 2& 7& 2\end{array}\right]$$

Note that $M$ is symmetric.