Simple polynomial equations over (2x2)-matrices
Submitted: 2025-06-26
|Accepted: 2025-09-01
|Published: 2025-12-03
Copyright (c) 2025 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 equation over matrices, matrix algebra, covering dimension
Supporting agencies:
The second named author was partially supported by NSERC Discovery Development Grant RGPIN-2015-06200
Abstract:
We consider the polynomial equation
Xn + an-1 ⋅ Xn-1 + ⋯ + a1 ⋅ X + a0 ⋅ I = O
over (2 x 2)-matrices X with the real entries, where I is the identity matrix, O is the null matrix, ai ∈ ℝ for each i and n ≥ 2 . We discuss its solution set S supplied with the natural Euclidean topology. We completely describe S. We also show that dim S =2.
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