An interlacing result for Hermitian matrices in Minkowski space

TL;DR

This paper extends the classical interlacing theorem to Hermitian matrices within Minkowski space, analyzing eigenvalue interlacing for a specific class of indefinite inner product matrices and their canonical forms.

Contribution

It introduces a new interlacing result for Hermitian matrices in Minkowski space and examines the canonical form and sign characteristic of the matrix pair.

Findings

Eigenvalue interlacing for Hermitian matrices in Minkowski space established

Canonical form and sign characteristic for the matrix pair analyzed

Results extend classical interlacing theorems to indefinite inner product spaces

Abstract

In this paper we will look at the well known interlacing problem, but here we consider the result for Hermitian matrices in the Minkowski space, an indefinite inner product space with one negative square. More specific, we consider the $n\times n$ matrix $A=\begin{bmatrix} J & u\\ -u^* & a\end{bmatrix}$ with $a\in\mathbb{R}$, $J=J^*$ and $u\in\mathbb{C}^{n-1}$. Then $A$ is $H$-selfadjoint with respect to the matrix $H=I_{n-1}\oplus(-1)$. The canonical form for the pair $(A,H)$ plays an important role and the sign characteristic coupled to the pair is also discussed.