On the determinantal conjecture of Marcus and de Oliveira
type: Item
instance of: scholarly article
title: On the determinantal conjecture of Marcus and de Oliveira
zbMATH Open document ID: 0768.15002
DOI: 10.1016/0024-3795(92)90319-6
author: B. Cload
author: S. W. Drury
published in: Linear Algebra and its Applications
publication date: 1993-05-16T00:00:00Z
review text: Let \(A\) and \(B\) be normal complex \(n\times n\) matrices with eigenvalues \(\lambda_ 1,\dots,\lambda_ n\) and \(\mu_ 1,\dots,\mu_ n\) respectively. \textit{M. Marcus} [Indiana Univ. Math. J. 22, 1137-1149 (1973; Zbl 0243.15025)] and \textit{G. N. de Oliveira} [Research problem: Normal matrices, Linear Multilinear Algebra 12, 153-154 (1982)] have conjectured that \(\text{det}(A + B) \in co\{\prod^ n_{j=1}(\lambda_ j + \mu_{\sigma(j)});\;\sigma \in S_ n\}\), where \(S_ n\) denotes the symmetric group on \(\{1,2,\dots,n\}\) and \(co\) denotes the convex hull in the complex plane. The authors provide a theorem and a counterexample relating to the determinantal conjecture of Marcus and de Oliveira.
reviewed by: Yueh-er Kuo
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 166218
zbMATH Keywords: normal complex matrices
zbMATH Keywords: eigenvalues
zbMATH Keywords: symmetric group
zbMATH Keywords: convex hull
zbMATH Keywords: determinantal conjecture
Wikidata QID: Q123099561
MaRDI profile type: Publication
full work available at URL: 0024-3795(92)90319-6
OpenAlex ID: W2077674425
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description: scientific article; zbMATH DE number 166218