Quadratic transformations on matrices: Rank preservers
type: Item
instance of: scholarly article
title: Quadratic transformations on matrices: Rank preservers
zbMATH Open document ID: 0839.15002
author: William Watkins
published in: Journal of Algebra
publication date: 1996-06-25T00:00:00Z
review text: An algebraically closed field \(F\) of characteristic not 2 and two \(n \times n\) matrices, one being \(X = (X_{ij})\) with independent indeterminates over \(F\) as entries, the second \(Q(X) = (q_{ij} (X))\) with quadratic \(F\)-polynomials in the \(X_{ij}\) as entries, are considered. It is shown that for \(n \geq 5\), \(Q(X)\) satisfies: \(\text{rank} (A^2) = r\) implies \(\text{rank} (Q(A)) = r\) for all \(A \in F^{n \times n}\) for \(r = 0\), 1, and 2, if and only if there exist invertible matrices \(P_1\), \(P_2\) in \(F^{n \times n}\) such that either \(Q(X) = P_1 X^2P_2\) or \(Q(X) = P_1 (X^2)^t P_2\), where \((X^2)^t\) denote the transpose of \((X^2)\).
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15A54
zbMATH DE Number: 843879
zbMATH Keywords: quadratic transformations on matrices
zbMATH Keywords: rank preservers
zbMATH Keywords: polynomial matrix
MaRDI profile type: Publication
full work available at URL: jabr.1996.0024
OpenAlex ID: W1980456641
DOI: 10.1006/JABR.1996.0024
description: scientific article; zbMATH DE number 843879