Definition of skew-symmetric matrix

Definition. A matrix $A$ is skew-symmetric (or antisymmetric) if $A^T = -A$.

Equivalently, $a_{ij} = -a_{ji}$ for all $i, j$. This implies:

• All diagonal entries are zero: $a_{ii} = 0$
• $A$ is always square

Example. $A = \begin{pmatrix} 0 & 3 & -1 \\ -3 & 0 & 2 \\ 1 & -2 & 0 \end{pmatrix}$ is skew-symmetric.

Properties:

• Every square matrix can be uniquely written as the sum of a symmetric and a skew-symmetric matrix: $A = \frac{1}{2}(A + A^T) + \frac{1}{2}(A - A^T)$
• The eigenvalues of a real skew-symmetric matrix are purely imaginary (or zero)
• For any vector $x$: $x^T A x = 0$