Block Tridiagonal Reduction of Perturbed Normal and Rank Structured Matrices

TL;DR

This paper extends properties of normal matrices to almost normal matrices satisfying specific quadratic equations, enabling structured reduction techniques for perturbed Hermitian and unitary matrices.

Contribution

It introduces generalizations of normal matrix properties to almost normal matrices with quadratic equations, aiding structured eigenvalue problem analysis.

Findings

Almost normal matrices can be reduced to block tridiagonal form.

The spectrum of such matrices lies on a straight line in the complex plane.

The methods apply to perturbed Hermitian and unitary matrices.

Abstract

It is well known that if a matrix $A\in\mathbb C^{n\times n}$ solves the matrix equation $f(A,A^H)=0$, where $f(x, y)$ is a linear bivariate polynomial, then $A$ is normal; $A$ and $A^H$ can be simultaneously reduced in a finite number of operations to tridiagonal form by a unitary congruence and, moreover, the spectrum of $A$ is located on a straight line in the complex plane. In this paper we present some generalizations of these properties for almost normal matrices which satisfy certain quadratic matrix equations arising in the study of structured eigenvalue problems for perturbed Hermitian and unitary matrices.