Self-interlacing polynomials II: Matrices with self-interlacing spectrum

TL;DR

This paper introduces a method to construct matrices with self-interlacing spectra, generalizing existing results and applying to bidiagonal and tridiagonal matrices, expanding understanding of eigenvalue distributions.

Contribution

It presents a new method for constructing sign definite matrices with self-interlacing spectra from totally nonnegative matrices, extending previous spectral results.

Findings

Method for constructing self-interlacing spectrum matrices from totally nonnegative matrices

Generalization of Holtz's result on real symmetric anti-bidiagonal matrices

Application to bidiagonal and tridiagonal matrices

Abstract

An $n\times n$ matrix is said to have a self-interlacing spectrum if its eigenvalues $\lambda_k$, $k=1,\ldots,n$, are distributed as follows $$ \lambda_1>-\lambda_2>\lambda_3>\cdots>(-1)^{n-1}\lambda_n>0. $$ A method for constructing sign definite matrices with self-interlacing spectra from totally nonnegative ones is presented. We apply this method to bidiagonal and tridiagonal matrices. In particular, we generalize a result by O. Holtz on the spectrum of real symmetric anti-bidiagonal matrices with positive nonzero entries.