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Homeskewsymmetric matrix
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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 $1\leq i\leq n,1\leq j\leq n$.
$A=\begin{pmatrix}a_{{11}}=0&\cdots&a_{{1n}}\\ \vdots&\ddots&\vdots\\ a_{{n1}}&\cdots&a_{{nn}}=0\end{pmatrix}$
The main diagonal entries are zero because $a_{{i,i}}=a_{{i,i}}$ implies $a_{{i,i}}=0$.
One can see 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. 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:

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

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