Quasi-Triangularization of Matrix Polynomials over Arbitrary Fields

TL;DR

This paper extends the quasi-triangularization of regular matrix polynomials to arbitrary fields, providing bounds on block sizes based on the Smith form factors, generalizing previous results over algebraically closed fields.

Contribution

It generalizes spectral quasi-triangularization results from algebraically closed and real fields to arbitrary fields, with bounds based on Smith form factors.

Findings

Spectral quasi-triangularization over arbitrary fields achieved.

Largest diagonal block size bounded by highest degree of irreducible factors.

Generalization of previous results over algebraically closed fields.

Abstract

In [19], Taslaman, Tisseur, and Zaballa show that any regular matrix polynomial $P(\lambda)$ over an algebraically closed field is spectrally equivalent to a triangular matrix polynomial of the same degree. When $P(\lambda)$ is real and regular, they also show that there is a real quasi-triangular matrix polynomial of the same degree that is spectrally equivalent to $P(\lambda)$, in which the diagonal blocks are of size at most $2 \times 2$. This paper generalizes these results to regular matrix polynomials $P(\lambda)$ over arbitrary fields $\mathbb{F}$, showing that any such $P(\lambda)$ can be quasi-triangularized to a spectrally equivalent matrix polynomial over $\mathbb{F}$ of the same degree, in which the largest diagonal block size is bounded by the highest degree appearing among all of the $\mathbb{F}$-irreducible factors in the Smith form for $P(\lambda)$.