Some further notes on the Marcus-de Oliveira determinantal conjecture
type: Item
instance of: scholarly article
title: Some further notes on the Marcus-de Oliveira determinantal conjecture
zbMATH Open document ID: 0797.15004
DOI: 10.1016/0024-3795(93)90140-J
author: Jorma K. Merikoski
author: Ari Virtanen
published in: Linear Algebra and its Applications
publication date: 1993-08-29T00:00:00Z
review text: Let \(A\) and \(B\) be normal matrices with eigenvalues \((a_ j)\) and \((b_ j)\), respectively. A famous conjecture claims that \(d=\text{det} (A+B)\) belongs to the convex hull of the products \(z_ \sigma = \prod (a_ j + b_{\sigma (j)})\), taken for all permutations \(\sigma\). The authors show that this is equivalent to the inequalities \(\min_ \sigma \text{Re} z_ \sigma \leq \text{Re} d \leq \max_ \sigma \text{Re} z_ \sigma \), and that this is true if all the \(z_ \sigma\)'s are collinear. They also show that the conjecture holds if among the \(a_ j\)'s (or, \(b_ j\)'s) there are at most two different values. The paper corrects an earlier paper of the same authors [Linear Algebra Appl. 121, 345-352 (1989; Zbl 0692.15003)].
Mathematics Subject Classification ID: 15A15
zbMATH DE Number: 399303
zbMATH Keywords: determinant of the sum of two normal matrices
zbMATH Keywords: eigenvalues
zbMATH Keywords: convex hull
zbMATH Keywords: inequalities
reviewed by: Jaroslav Zemánek
MaRDI profile type: Publication
Wikidata QID: Q123308753
cites work: The determinant of the sum of two normal matrices with prescribed eigenvalues
cites work: On the determinantal conjecture of Marcus and de Oliveira
cites work: Q5654173
cites work: Some notes on de Oliveira's determinantal conjecture
full work available at URL: 0024-3795(93)90140-j
OpenAlex ID: W2033253657
description: scientific article; zbMATH DE number 399303