de Seguins Pazzis, Clément The sum and the product of two quadratic matrices: regular cases. (English) Zbl 1497.15013 Adv. Appl. Clifford Algebr. 32, No. 5, Paper No. 54, 43 p. (2022). A matrix is called quadratic if it satisfies a quadratic equation. Let \(p(t)\) and \(q(t)\) be two quadratic polynomials over an arbitrary field \(\mathbb{F}\) and let \(\mathrm{Root}(p)\) and \(\mathrm{Root}(q) \) denote their sets of roots. Let \(M_{n}(\mathbb{F})\) be the algebra of \( n\times n\) matrices over \(\mathbb{F}.\) The \((p,q)\)-difference problem considered here is: characterise the matrices \(C\in M_{n}(\mathbb{F})\) such that there exist \(A,B\in M_{n}(\mathbb{F})\) with \(p(A)=q(B)=0\) and \( C=A-B\). Similarly, if \(p(0)q(0)\neq 0\) then the \((p,q)\)-quotient problem is to characterise the \(C\) such that \(C=AB^{-1}\). Previously, only special problems of this type have been considered, see, e.g., [J.-H. Wang, Linear Algebra Appl. 229, 127–149 (1995; Zbl 0833.15011)]. The present paper gives a complete solution for the regular cases; namely, when no eigenvalue of \(C\) lies in \(\mathrm{Root}(p)- \mathrm{Root}(q)\) for the \((p,q)\)-difference problem and no eigenvalue of \(C\) lies in \(\mathrm{Root}(p)\mathrm{Root}(q)^{-1}\) for the \((p,q)\)-quotient problem. In these cases, the matrices \(C\) are similar to direct sums of companion matrices of specified types depending only on \(p\) and \(q\). The paper also includes a complete description of the solution for the exceptional cases but proofs are deferred to a later paper. An important role in the solution is played by a \(4\)-dimensional quaternion algebra \( \mathcal{W(}p,q,x)\). Reviewer: John D. Dixon (Ottawa) Cited in 2 ReviewsCited in 4 Documents MSC: 15A21 Canonical forms, reductions, classification 15B33 Matrices over special rings (quaternions, finite fields, etc.) 15A29 Inverse problems in linear algebra 11R52 Quaternion and other division algebras: arithmetic, zeta functions Keywords:quadratic matrices; rational canonical form; companion matrices; quaternion algebras Citations:Zbl 0833.15011 PDFBibTeX XMLCite \textit{C. de Seguins Pazzis}, Adv. Appl. Clifford Algebr. 32, No. 5, Paper No. 54, 43 p. 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