The determinant of the sum of two normal matrices with prescribed eigenvalues
type: Item
instance of: scholarly article
title: The determinant of the sum of two normal matrices with prescribed eigenvalues
zbMATH Open document ID: 0576.15006
DOI: 10.1016/0024-3795(85)90231-9
author: Natalia Bebiano
author: João Filipe Queiró
published in: Linear Algebra and its Applications
publication date: 1985-01-01T00:00:00Z
review text: This paper presents some partial results on a conjecture of the present reviewer: let \(A\) and \(B\) be two \(n\times n\) normal matrices with eigenvalues \(\alpha_ 1,\ldots,\alpha_ n\) and \(\beta_ 1,\ldots,\beta_ n\), respectively. Then the determinant of \(A+B\) belongs to the convex hull of the points \(\prod_{i}(\alpha_ i+\beta_{\sigma}+\beta_{\sigma (i)})\), \(\sigma\) running over the symmetric group \(S_ n\) of degree \(n\). The deepest result is perhaps the following. Let \(\Delta\) be the set described by the determinant of \(A+B\) when \(A\) and \(B\) run over all the normal matrices which have the above eigenvalues and denote by \(\partial \Delta\) its boundary. If \(z\) belongs to \(\partial \Delta\) and is different from zero and if in a neighborhood of \(z\), \(\Delta\) is contained in an angle with vertex at \(z\) and measuring less than \(\pi\), then there is a \(\sigma\) in \(S_ n\) such that \[ z=\prod_{i}(\alpha_ j+\beta_{\sigma (j)}). \]
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 3920681
zbMATH Keywords: prescribed eigenvalues
zbMATH Keywords: normal matrices
zbMATH Keywords: determinant
MaRDI profile type: Publication
full work available at URL: 0024-3795(85)90231-9
OpenAlex ID: W2056516742
cites work: Bounds for the Determinant of the Sum of Hermitian Matrices
cites work: Elementary inclusion relations for generalized numerical ranges
cites work: Normality and the Higher Numerical Range
cites work: On the shape of the generalized numerical ranges
cites work: On eigenvalues of sums of normal matrices
description: scientific article; zbMATH DE number 3920681