Simple polynomial equations over (m x m)-matrices
Submitted: 2026-02-24
|Accepted: 2026-04-26
|Published: 2026-06-03
Copyright (c) 2026 Vitalij Chatyrko, Alexandre Karassev
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
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Keywords:
Polynomial equations over matrices, Matrix algebra, Covering dimension
Supporting agencies:
Natural Sciences and Engineering Research Council of Canada (NSERC) RGPIN-2015-06200
Abstract:
Let m be any integer ≥ 3 . We consider the polynomial equation
Xn+an-1·Xn-1+···+a1·X+a0·I=O,
over (m x m)-matrices X with the real entries, where I is the identity matrix, O is the null matrix, ai ∈ ℝ for each i and n ≥ 1. We discuss its solution set S supplied with the natural Euclidean topology.
In particular, we describe the solution set S for m=3 and calculate its dimension.
References:
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