Interlacing properties of the eigenvalues of some matrix classes

TL;DR

This paper proves eigenvalue interlacing properties for Kotelyansky matrices and their generalizations, establishing inequalities for eigenvalues and expanding understanding of matrix spectral behavior.

Contribution

It introduces eigenvalue interlacing results for Kotelyansky matrices and their generalizations, extending classical interlacing theory to new matrix classes.

Findings

Eigenvalue interlacing property established for Kotelyansky matrices

Interlacing inequalities for other eigenvalues proved

Generalizations of Kotelyansky matrices also exhibit interlacing

Abstract

We establish the eigenvalue interlacing property (i.e. the smallest real eigenvalue of a matrix is less than the smallest real eigenvalue of any its principal submatrix) for the class of matrices, introduced by Kotelyansky (all principal and all almost principal minors of these matrices are positive). We show that certain generalizations of Kotelyansky and totally positive matrices also possess this property. We prove some interlacing inequalities for the other eigenvalues of Kotelyansky matrices.