About the matrix function X->AX+XA

2012/10/02 by Gérald Bourgeois, Gerald Bourgeois, Bourgeois, Gerald
Computer Science · Engineering · Mathematics · #15A15 #15A21 #Advanced Topics in Algebra #FOS: Mathematics #Matrix Theory and Algorithms #Rings and Algebras (math.RA) #graph theory and CDMA systems #math.RA #msc:15A15 #msc:15A21

paper · pdf · doi:10.48550/arxiv.1210.0766

7 pages

arxiv created 2012/10/02 · openalex publication_date 2012/10/02 · arxiv updated 2012/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be an infinite field such that its characteristic is not 2. We show that, for every A\inM_n(K) such that rank(A)≥ n/2, there exists B\inM_n(K) such that B is similar to A and A+B is invertible. Let K be a subfield of ℝ. We show that, if n is even, then for every X\inM_n(K), det(AX+XA)≥ 0 if and only if either rank(A)<n/2 or there exists α∈ K,α≤ 0, such that A²=αI_n.

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