# 1.1.4 Transpose matrix

Transposing a matrix means that the rows become the columns and the columns become the rows, to an n by m matrix becomes m by n

$$
A = \begin{bmatrix}
a_{11} & a_{12} & \ldots & a_{1n}\\
a_{21} & a_{22} & \ldots & a_{2n}\\
\vdots & \vdots & \ddots & \vdots\\
a_{m1} & a_{m2} & \ldots & a_{mn}\\
\end{bmatrix}
$$

$$
A^T = \begin{bmatrix}
a_{11} & a_{21} & \ldots & a_{m1}\\
a_{12} & a_{22} & \ldots & a_{m2}\\
\vdots & \vdots & \ddots & \vdots\\
a_{1n} & a_{2n} & \ldots & a_{mn}\\
\end{bmatrix}
$$

Writing this in terms of matrix elements is simply $A_{ij} = A_{ji}$