Transpose, Symmetric and Skew-Symmetric Matrices

Transpose. The transpose of a matrix $A$, written $A^{\mathsf{T}}$ (or $A'$), is obtained by interchanging its rows and columns — the first row becomes the first column, the second row the second column, and so on. If $A$ is of order $m\times n$, then $A^{\mathsf{T}}$ is of order $n\times m$. For example, $A=\begin{pmatrix} 2 & -1 & 5 \\ 0 & 3 & 4 \end{pmatrix}\ \Rightarrow\ A^{\mathsf{T}}=\begin{pmatrix} 2 & 0 \\ -1 & 3 \\ 5 & 4 \end{pmatrix}.$

The transpose obeys three rules used throughout the chapter: $(A^{\mathsf{T}})^{\mathsf{T}}=A$; $(A+B)^{\mathsf{T}}=A^{\mathsf{T}}+B^{\mathsf{T}}$; and, most important, the reversal law $(AB)^{\mathsf{T}}=B^{\mathsf{T}}A^{\mathsf{T}}$ — the transpose of a product is the product of the transposes in the reverse order.

Symmetric matrix. A square matrix $A$ is symmetric if it equals its own transpose: $A^{\mathsf{T}}=A,\quad\text{i.e. } a_{ij}=a_{ji}\ \text{for all } i,j.$ Its entries are mirror images across the leading diagonal, e.g. $\begin{pmatrix} 1 & 5 & 3 \\ 5 & 2 & 6 \\ 3 & 6 & 4 \end{pmatrix}$.

Skew-symmetric matrix. A square matrix $A$ is skew-symmetric (or anti-symmetric) if $A^{\mathsf{T}}=-A,\quad\text{i.e. } a_{ij}=-a_{ji}\ \text{for all } i,j.$ Setting $i=j$ forces $a_{ii}=-a_{ii}$, so every diagonal element of a skew-symmetric matrix is zero, e.g. $\begin{pmatrix} 0 & 2 & -3 \\ -2 & 0 & 4 \\ 3 & -4 & 0 \end{pmatrix}$. …