Skew-symmetric matrix

Definition

Let $R$ be a ring and let $A = (a_{ij})$ be an $n \times n$ square matrix over $R$. The matrix $A$ is skew-symmetric when $A^T = -A$, where $A^T$ is the transpose of $A$. Equivalently, $a_{ji} = -a_{ij}$ for all $i$ and $j$.

Remarks

If $2$ is invertible in $R$, for example over $\R$, then each diagonal entry of a skew-symmetric matrix is $0$. Over a field of characteristic not $2$, each square matrix is the sum of a symmetric matrix and a skew-symmetric matrix: $A = \tfrac12 (A + A^T) + \tfrac12 (A - A^T)$. A real skew-symmetric matrix of odd order has determinant $0$. The matrix $\begin{pmatrix} 0 & 2 \\ -2 & 0 \end{pmatrix}$ is skew-symmetric.

Explanations

L1 human

In a skew-symmetric matrix, the entry across the diagonal is the negative of the entry here. Over the real numbers this forces zeros on the diagonal, because only $0$ equals its own negative. These matrices describe infinitesimal rotations.