On the canonical structure of regular pencil of singular matrix-functions

TL;DR

This paper investigates the canonical structure of regular pencils of matrix-functions with singular matrices, focusing on cases where determinants vanish and roots of the characteristic equation have constant multiplicity.

Contribution

It provides a detailed analysis of the canonical form of such matrix pencils under specific singularity and multiplicity conditions.

Findings

Canonical structure characterized for regular singular matrix pencils

Conditions for constant multiplicity of roots established

Framework for analyzing singular matrix-function pencils developed

Abstract

The work is devoted to investigation of the canonical structure of regular, in domain of definition ${\cal U}\subseteq {\mathbb R}^{m}$, the pencil of matrix-functions $A(x)+\lambda B(x)$. It is supposed that $\det A(x)\equiv 0$ and $\det B(x)\equiv 0\;\; \forall \ x \in {\cal U}$, and all roots of the characteristic equation $\det(A(x)+\lambda B(x))=0$ with $\lambda=\lambda (x)\in C^{p}({\cal U})$, are of constant multiplicity.