View source for Skew-symmetric matrix

A square [[matrix]] $A$ over a field of characteristic $\ne 2$ such that $A^T = -A$. The rank of a skew-symmetric matrix is an even number. Any square matrix $B$ over a field of characteristic $\ne 2$ is the sum of a [[symmetric matrix]] and a skew-symmetric matrix:
$$
B = \frac12(B + B^T) + \frac12(B - B^T) \ .
$$
The non-zero roots of the [[characteristic polynomial]] of a real skew-symmetric matrix are purely [[imaginary number]]s. A real skew-symmetric matrix is [[Similar matrices|similar]] to a matrix
$$
\text{diag}[A_1,A_2,\ldots,A_t,0,0,\ldots]
$$
where
$$
A_i = \alpha_i \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right)
$$
with $\alpha_i$ real numbers, $i = 1,\ldots,t$. The [[Jordan normal form]] $J$ of a complex skew-symmetric matrix possesses the following properties: 1) a Jordan block $J_m(\lambda)$ with elementary divisor $(X-\lambda)^m$, where $\lambda \ne 0$, is repeated in $J$ as many times as is the cell $J_m(-\lambda)$; and 2) if $m$ is even, the Jordan block $J_m(0)$ with elementary divisor $X^m$ is repeated in $J$ an even number of times. Any complex Jordan matrix with the properties 1) and 2) is similar to some skew-symmetric matrix.

The set of all skew-symmetric matrices of order $n$ over a field $k$ forms a [[Lie algebra]] over $k$ with respect to matrix addition and the commutator $[A,B] = AB - BA$.

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  F.R. [F.R. Gantmakher] Gantmacher,   "The theory of matrices" , '''1''' , Chelsea, reprint  (1977)  (Translated from Russian)</TD></TR>
</table>



====Comments====
The Lie algebra of skew-symmetric matrices over a field $k$ of size $n \times n$ is denoted by $\mathfrak{so}(n,k)$. The complex Lie algebras $\mathfrak{so}(2n,\mathbf{C})$ ($n \ge 4$) and $\mathfrak{so}(2n_1,\mathbf{C})$ ($n \ge 2$) are simple of type $D_n$ and $B_n$, respectively.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Helgason,   "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press  (1978)  pp. Chapt. X</TD></TR>
</table>

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