Symmetric and Skew-Symmetric Matrices

The two definitions

A square matrix $A$ is symmetric if $A^T = A$, and skew-symmetric if $A^T = -A$. Any square matrix can be split into a symmetric part plus a skew-symmetric part:

Worked example

Express $A = \begin{bmatrix} 3 & 5 \\ 1 & -1 \end{bmatrix}$ as the sum of a symmetric and a skew-symmetric matrix.

Solution: First find $A^T = \begin{bmatrix} 3 & 1 \\ 5 & -1 \end{bmatrix}$.

Checking the two pieces: the first matrix is symmetric (it equals its own transpose), the second is skew-symmetric (its transpose is its negative), and adding them back together gives exactly $A$.