skewsymmetric matrix
Definition:
Let $A$ be an square matrix^{} of
order $n$ with real entries $({a}_{ij})$.
The matrix $A$ is skewsymmetric if ${a}_{ij}={a}_{ji}$ for all $1\le i\le n,1\le j\le n$.
$A=\left(\begin{array}{ccc}\hfill {a}_{11}=0\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{1n}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill {a}_{n1}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{nn}=0\hfill \end{array}\right)$
The main diagonal entries are zero because ${a}_{i,i}={a}_{i,i}$ implies ${a}_{i,i}=0$.
One can see skewsymmetric matrices as a special case of complex skewHermitian matrices. Thus, all properties of skewHermitian matrices also hold for skewsymmetric matrices.
Properties:

1.
The matrix $A$ is skewsymmetric if and only if ${A}^{t}=A$, where ${A}^{t}$ is the matrix transpose

2.
For the trace operator, we have that $\mathrm{tr}(A)=\mathrm{tr}({A}^{t})$. Combining this with property (1), it follows that $\mathrm{tr}(A)=0$ for a skewsymmetric matrix $A$.

3.
Skewsymmetric matrices form a vector space^{}: If $A$ and $B$ are skewsymmetric and $\alpha ,\beta \in \mathbb{R}$, then $\alpha A+\beta B$ is also skewsymmetric.

4.
Suppose $A$ is a skewsymmetric matrix and $B$ is a matrix of same order as $A$. Then ${B}^{t}AB$ is skewsymmetric.

5.
All eigenvalues^{} of skewsymmetric matrices are purely imaginary or zero. This result is proven on the page for skewHermitian matrices.

6.
According to Jacobi’s Theorem, the determinant^{} of a skewsymmetric matrix of odd order is zero.
Examples:

•
$\left(\begin{array}{cc}\hfill 0\hfill & \hfill b\hfill \\ \hfill b\hfill & \hfill 0\hfill \end{array}\right)$

•
$\left(\begin{array}{ccc}\hfill 0\hfill & \hfill b\hfill & \hfill c\hfill \\ \hfill b\hfill & \hfill 0\hfill & \hfill e\hfill \\ \hfill c\hfill & \hfill e\hfill & \hfill 0\hfill \end{array}\right)$
Title  skewsymmetric matrix 

Canonical name  SkewsymmetricMatrix 
Date of creation  20130322 12:01:05 
Last modified on  20130322 12:01:05 
Owner  Daume (40) 
Last modified by  Daume (40) 
Numerical id  10 
Author  Daume (40) 
Entry type  Definition 
Classification  msc 1500 
Related topic  SelfDual 
Related topic  AntiSymmetric 
Related topic  SkewHermitianMatrix 
Related topic  AntisymmetricMapping 