Factorizations into Normal Matrices in Indefinite Inner Product Spaces

TL;DR

This paper demonstrates that any nonsingular matrix in an indefinite inner product space can be factorized into a product of at most three specialized normal matrices, expanding the understanding of matrix factorizations in such spaces.

Contribution

It introduces a novel factorization of nonsingular matrices into three normal matrices with specific properties in indefinite inner product spaces.

Findings

Any nonsingular matrix can be factorized into three normal matrices.

One factor is unitary, another is selfadjoint with eigenvalues in the right half-plane.

The third is a normal involutory matrix with a neutral negative eigenspace.

Abstract

We show that any nonsingular (real or complex) square matrix can be factorized into a product of at most three normal matrices, one of which is unitary, another selfadjoint with eigenvalues in the open right half-plane, and the third one is normal involutory with a neutral negative eigenspace (we call the latter matrices normal neutral involutory). Here the words normal, unitary, selfadjoint and neutral are understood with respect to an indefinite inner product.