PlanetMath: skew-symmetric matrix

(more info)

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skew-symmetric matrix

(Definition)

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

$A = \begin{pmatrix} a_{11}=0 & \cdots & a_{1n} \ \vdots & \ddots & \vdots \ a_{n1} & \cdots & a_{nn}=0 \end{pmatrix}$

The main diagonal entries are zero because $a_{i,i} = -a_{i,i}$ implies $a_{i,i} = 0$ .

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:

The matrix is skew-symmetric if and only if , where is the matrix transpose
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 .
Skew-symmetric matrices form a vector space: If and are skew-symmetric and $\alpha, \beta\in \mathbb{R}$ , then $\alpha A + \beta B$ is also skew-symmetric.
Suppose is a skew-symmetric matrix and is a matrix of same order as . Then is skew-symmetric.
All eigenvalues of skew-symmetric matrices are purely imaginary or zero. This result is proven on the page for skew-Hermitian matrices.
According to Jacobi's Theorem, the determinant of a skew-symmetric matrix of odd order is zero.