Authors: | Dinčić, Nebojša Đorđević, Bogdan |
Affiliations: | Mathematics Mathematical Institute of the Serbian Academy of Sciences and Arts |
Title: | On the intrinsic structure of the solution set to the Yang–Baxter-like matrix equation | Journal: | Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas | Volume: | 116 | Issue: | 2 | First page: | 73 | Issue Date: | 1-Apr-2022 | Rank: | ~M21a | ISSN: | 1578-7303 | DOI: | 10.1007/s13398-022-01214-8 | Abstract: | In this paper we analyze isolated and connected points of the solution set to the Yang–Baxter-like matrix equation AXA= XAX. In particular, if A is a regular matrix, we prove that the trivial solution X= 0 is always isolated in the solution set, while the trivial solution X= A is isolated under some natural conditions, and an example shows that these conditions cannot be omitted. Conversely, we demonstrate that the two trivial solutions can be path-connected in the solution set when A is singular. Furhter, we prove that every nontrivial non-commuting solution is always contained in some path-connected subset of the solution set, regardless of whether A is regular or singular. Additionally, we develop new methods for obtaining infinitely many new nontrivial non-commuting solutions (for both regular and singular A). Explicit examples are provided after almost every theoretical result. |
Keywords: | Generalized inverses | Nonlinear matrix equations | Sylvester equation | Yang–Baxter-like matrix equation | Publisher: | Springer Link |
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