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 numbers. A real skew-symmetric matrix is 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

[1]	F.R. [F.R. Gantmakher] Gantmacher, "The theory of matrices" , 1 , Chelsea, reprint (1977) (Translated from Russian)

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

[a1]	S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) pp. Chapt. X