The transpose of a matrix

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There are exercises as selftest for this article.

The transpose of an m-by-n matrix $\mathbf{A}$ is the n-by-m matrix $\mathbf{A}^T$ where all the colums are written as rows and all the rows as colums:

$\mathbf{A}= \begin{pmatrix} a_{11} & a_{12} & \dots &a_{1n} \\ a_{21} &\ddots & &\vdots\\ \vdots & & & \\ a_{m1} & \dots & &a_{mn} \end{pmatrix}, \quad \mathbf{A}^T = \begin{pmatrix} a_{11} & a_{21} & \dots &a_{m1} \\ a_{12} &\ddots & &\vdots\\ \vdots & & & \\ a_{1n} & \dots & &a_{mn} \end{pmatrix}$

If an n-by-n square matrix is concerned the transpose is obtained by mirroring the matrix at its first diagonal from top left to bottom right.

In general the following arithmetic rules hold:

$\begin{align} (\mathbf{A}+\mathbf{B})^T &= \mathbf{A}^T + \mathbf{B}^T\\ (c \cdot \mathbf{A})^T &= c \cdot \mathbf{A}^T\\ \left(\mathbf{A}^T\right)^T &= \mathbf{A}\\ (\mathbf{A} \cdot \mathbf{B})^T &= \mathbf{B}^T \cdot \mathbf{A}^T\\ \left(\mathbf{A}^{-1}\right)^T &= \left(\mathbf{A}^T\right)^{-1} \end{align}$

Example: The transpose of a matrix $\mathbf{A}= \begin{pmatrix} 2 & 3 & 0 \\ 1 & 4 & 2 \\ 5 & 6 & 2 \end{pmatrix}, \quad \mathbf{A}^T = \begin{pmatrix} 2 & 1 & 5 \\ 3 & 4 & 6 \\ 0 & 2 & 2 \end{pmatrix}$