Definition:
Let $A$ be an square matrix of order $n$ with real entries $(a_{ij})$ . The matrix $A$ is skew-symmetric if $a_{ij} = -a_{ji}$ for all $1 \leq i \leq n, 1 \leq j \leq n$ .

One can see skew-symmetric matrices as a special case of complex skew-Hermitian matrices. Thus, all properties of skew-Hermitian matrices also hold for skew-symmetric matrices.

Properties:

1. The matrix $A$ is skew-symmetric if and only if $A^t = -A$ , where $A^t$ is the matrix transpose
2. For the trace operator, we have that $\operatorname{tr}(A) = \operatorname{tr}(A^t)$ . Combining this with property (1), it follows that $ \operatorname{tr}(A)=0$ for a skew-symmetric matrix $A$ .
3. Skew-symmetric matrices form a vector space: If $A$ and $B$ are skew-symmetric and $\alpha, \beta\in \mathbb{R}$ , then $\alpha A + \beta B$ is also skew-symmetric.
4. Suppose $A$ is a skew-symmetric matrix and $B$ is a matrix of same order as $A$ . Then $B^t A B$ is skew-symmetric.
5. All eigenvalues of skew-symmetric matrices are purely imaginary or zero. This result is proven on the page for skew-Hermitian matrices.
6. According to Jacobi's Theorem, the determinant of a skew-symmetric matrix of odd order is zero.

Examples:

• $\begin{pmatrix} 0 & b \\ -b & 0 \end{pmatrix}$
• $\begin{pmatrix} 0 & b & c \\ -b & 0 & e \\ -c & -e & 0 \end{pmatrix}$