Symmetric and Skew-Symmetric Matrices

A square matrix $A$ is:

• symmetric if $A^T=A$, i.e. $a_{ij}=a_{ji}$ for every $i,j$ — the matrix mirrors itself across the main diagonal;
• skew-symmetric if $A^T=-A$, i.e. $a_{ij}=-a_{ji}$ for every $i,j$. Putting $i=j$ gives $a_{ii}=-a_{ii}$, i.e. $2a_{ii}=0$, so every diagonal entry of a skew-symmetric matrix must be $0$.

A matrix that is both symmetric and skew-symmetric simultaneously must be the zero matrix (since $A=A^T=-A \Rightarrow 2A=O$).

Theorem 7.1. For any square matrix $A$ with real entries, $A+A^T$ is symmetric and $A-A^T$ is skew-symmetric.

Proof. Let $B=A+A^T$. Using transpose properties 1 and 3: $B^T=(A+A^T)^T=A^T+(A^T)^T=A^T+A=B$, so $B^T=B$, i.e. $B$ is symmetric. Now let $C=A-A^T$. Then $C^T=A^T-(A^T)^T=A^T-A=-(A-A^T)=-C$, so $C$ is skew-symmetric. $\blacksquare$

Theorem 7.2 (decomposition). Every square matrix $A$ can be written as a sum of a symmetric matrix and a skew-symmetric matrix:

Proof. The right side clearly adds up to $A$. By Theorem 7.1, $A+A^T$ is symmetric and $A-A^T$ is skew-symmetric; since $(kA)^T=kA^T$, scaling by the constant $\tfrac12$ preserves symmetry/antisymmetry, so $\tfrac12(A+A^T)$ is symmetric and $\tfrac12(A-A^T)$ is skew-symmetric. $\blacksquare$ …