The antitriangular factorization of skew-symmetric matrices

TL;DR

This paper introduces algorithms for transforming skew-symmetric matrices into antitriangular forms using orthogonal similarity transformations, revealing structural similarities with symmetric and skew-Hermitian matrices.

Contribution

It develops new algorithms for antitriangular factorization of skew-symmetric matrices and shows structural parallels with symmetric and skew-Hermitian matrices.

Findings

Antitriangular form always obtained for skew-symmetric matrices.

Two-sided permutation transforms the form into a multi-arrowhead matrix.

Structural similarity between skew-Hermitian and symmetric matrices' forms.

Abstract

In this paper we develop algorithms for orthogonal similarity transformations of skew-symmetric matrices to simpler forms. The first algorithm is similar to the algorithm for the block antitriangular factorization of symmetric matrices, but in the case of skew-symmetric matrices, an antitriangular form is always obtained. Moreover, a simple two-sided permutation of the antitriangular form transforms the matrix into a multi-arrowhead matrix. In addition, we show that the block antitriangular form of the skew-Hermitian matrices has the same structure as the block antitriangular form of the symmetric matrices.