On the invariant polynomials of Jordan products
type: Item
instance of: scholarly article
title: On the invariant polynomials of Jordan products
zbMATH Open document ID: 1020.15017
DOI: 10.1016/S0024-3795(02)00448-2
author: Enide Andrade Martins
author: F. C. Silva
published in: Linear Algebra and its Applications
publication date: 2003-03-11T00:00:00Z
review text: Consider the ring \(F^{n\times n}\) of all \(n\times n\) matrices over an algebraically closed field \(F\) of characteristic \(\neq 2\). For each \(A\in F^{n\times n}\) define \(R_{F}(A)\) to be the minimum of \text{rank}\((A-\lambda I)\) for \(\lambda\in F\). It is known [see \textit{G. N. Oliveira, E. M. Sá} and \textit{J. A. Dias de Silva}, Linear Multilinear Algebra 5, 119-128 (1977; Zbl 0367.15010)] that \(n-R_{F}(A)\) is equal to the number of nonconstant invariant factors of \(A\). In the present paper the authors prove that, for each \(A\in F^{n\times n}\), the value of \(R_{F}(XA+AX)\) runs over \(0,1,\dots ,2 \text{rank}(A)\) as \(X\) runs over \(F^{n\times n}\). They also prove a similar, but more complicated, theorem (involving the degrees of the invariant factors of \(A)\) for the values of \(R_{F}(XA+AX)\) when \(X\) is restricted to the invertible matrices in \(F^{n\times n}\).
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 15A21
zbMATH DE Number: 1880033
zbMATH Keywords: invariant polynomials
zbMATH Keywords: Jordan products
zbMATH Keywords: matrix ring
zbMATH Keywords: rank
zbMATH Keywords: invariant factors
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1880033