Transpose of a Matrix; Symmetric and Skew-Symmetric Matrices

Transpose of a matrix

The transpose of a matrix $A$, written $A^T$ (or $A'$), is obtained by interchanging its rows and columns — the first row of $A$ becomes the first column of $A^T$, and so on. If $A$ is of order $m\times n$, then $A^T$ is of order $n\times m$. For example,

Taking the transpose twice returns the original matrix: $(A^T)^T=A$, always.

Symmetric matrix

A SQUARE matrix $A$ is symmetric if $A^T=A$ — that is, every element reflected across the leading diagonal is identical to the element it is reflected onto ($a_{ij}=a_{ji}$ for every $i,j$). Symmetric matrices arise naturally, for instance, in a table of pairwise distances or correlations between the same set of items.

Skew-symmetric matrix

A SQUARE matrix $A$ is skew-symmetric if $A^T=-A$ — that is, $a_{ij}=-a_{ji}$ for every $i,j$. Setting $i=j$ forces $a_{ii}=-a_{ii}$, so EVERY diagonal element of a skew-symmetric matrix must be $0$.

Every square matrix splits into a symmetric part and a skew-symmetric part

For any square matrix $A$,

$A=\underbrace{\tfrac{1}{2}(A+A^T)}_{\text{symmetric}}+\underbrace{\tfrac{1}{2}(A-A^T)}_{\text{skew-symmetric}}$ …