skew-symmetric matrix

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 $1\leq i\leq n,1\leq j\leq n$.

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

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:

1. 1.

The matrix $A$ is skew-symmetric if and only if $A^{t}=-A$, where $A^{t}$ is the matrix transpose

2. 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. 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. 4.

Suppose $A$ is a skew-symmetric matrix and $B$ is a matrix of same order as $A$. Then $B^{t}AB$ is skew-symmetric.

5. 5.

All eigenvalues of skew-symmetric matrices are purely imaginary or zero. This result is proven on the page for skew-Hermitian matrices.

6. 6.

According to Jacobi’s Theorem, the determinant of a skew-symmetric matrix of odd order is zero.

Examples:

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

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

Title skew-symmetric matrix SkewsymmetricMatrix 2013-03-22 12:01:05 2013-03-22 12:01:05 Daume (40) Daume (40) 10 Daume (40) Definition msc 15-00 SelfDual AntiSymmetric SkewHermitianMatrix AntisymmetricMapping