Matrices satisfying a conjecture of G. N. de Oliveira on determinants
type: Item
instance of: scholarly article
title: Matrices satisfying a conjecture of G. N. de Oliveira on determinants
zbMATH Open document ID: 0592.15003
DOI: 10.1016/0024-3795(86)90023-6
author: João Da Providência
author: Natalia Bebiano
published in: Linear Algebra and its Applications
publication date: 1986-01-01T00:00:00Z
review text: Let \(A\) and \(B\) be normal \(n\times n\) matrices with eigenvalues \(\alpha_ 1,...,\alpha_ n\) and \(\beta_ 1,...,\beta_ n\) respectively. Once the present reviewer [Normal matrices (research problem), Linear Multilinear Algebra 12, 153-154 (1982)] conjectured that if \(U\) is a unitary matrix then the determinant of \(A+UBU^*\) belongs to the convex hull of the \(n!\) points \(\prod^{n}_{j=1}(\alpha_ j+\beta_{\sigma(j)}\), \(\sigma\) running over the symmetric group of degree n. In the present paper the authors prove that the conjecture is true if \(A\) is positive definite Hermitian and \(B\) is a skew-Hermitian matrix. It should be pointed out that the proof of the result in this paper is quite complicated. The conjecture is known to be true also in the case \(A\) and \(B\) are Hermitian. In fact it was this result of \textit{M. Fiedler} [Proc. Amer. Math. Soc. 30, 27-31 (1971; Zbl 0277.15010)] that inspired the conjecture. The proof of Fiedler's result is difficult. So it seems that a complete answer to the conjecture will also be rather difficult.
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 3950730
zbMATH Keywords: normal matrices
zbMATH Keywords: eigenvalues
zbMATH Keywords: determinant
zbMATH Keywords: skew-Hermitian matrix
MaRDI profile type: Publication
full work available at URL: 0024-3795(86)90023-6
OpenAlex ID: W2142974529
Wikidata QID: Q122941373
cites work: The determinant of the sum of two normal matrices with prescribed eigenvalues
cites work: Bounds for the Determinant of the Sum of Hermitian Matrices
cites work: Q5547252
cites work: Q5338702
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description: scientific article; zbMATH DE number 3950730