Multiple Factorization of Skew-Symmetric Matrices

Abstract

Multiple factorization $A=\Pi D\Pi^{\textrm{T}}$ is proposed for real skew-symmetric matrix $A\neq 0$ of order $n\geq 3$. The block-diagonal factor $D$ contains skew-symmetric invertible blocks of order 2 and the zero block of order $n-\textrm{rank}A$ if $\textrm{rank}A<n$. The matrix $\Pi$ is an alternate product of permutation matrices and unit lower triangular matrices with two columns. The applied approach is economical and contributes to the computational stability. The number of arithmetic operations $\sim{\frac{1}{3}}n^{3}$. The inverse matrix $A^{-1}$ and the skeletal decomposition of $A$ are presented in factorized form without additional calculations.