Quasi-triangularization of matrix polynomials over arbitrary fields

Anguas, Luis M. and Dopico, Froilán M. and Hollister, Richard and Mackey, D. Steven (2021) Quasi-triangularization of matrix polynomials over arbitrary fields. [MIMS Preprint]

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Abstract

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

Item Type:	MIMS Preprint
Uncontrolled Keywords:	matrix polynomials, triangularization, arbitrary field, majorization, inverse problem, Mobius transformation
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 05 Combinatorics MSC 2010, the AMS's Mathematics Subject Classification > 12 Field theory and polynomials MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
Depositing User:	Dr. D. Steven Mackey
Date Deposited:	09 Dec 2021 08:52
Last Modified:	09 Dec 2021 08:52
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/2839

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