An approach to the permanental-dominance conjecture
type: Item
instance of: scholarly article
title: An approach to the permanental-dominance conjecture
zbMATH Open document ID: 0835.15004
DOI: 10.1016/0024-3795(94)90117-1
author: George W. Soules
published in: Linear Algebra and its Applications
publication date: 1994-08-15T00:00:00Z
review text: Let \(A = (a_{ij})\) denote a complex \(n\)-square matrix. Then the Schur power matrix of \(A\) is the \(n!\)-square matrix \(\prod_A : = (\prod^n_{i = 1} a_{\sigma (i), \tau (i)})\), where \(\sigma, \tau \in S_n\) (the full symmetric group of order \(n)\). The maximum eigenvalue conjecture (MEC) states that, if \(A\) is positive semidefinite Hermitian, then \(\text{per} (A)\) is the largest eigenvalue of \(\prod_A\). It is noteworthy that the MEC is stronger than the well-known permanental dominance conjecture (PDC). This says that, if \(G\) is a subgroup of \(S_n\), and if \(\chi\) is any character of \(G\), then, for any positive semidefinite Hermitian \(n\)-square matrix \(A = (a_{ij})\), the generalized matrix function \[ d^G_\chi (A) = {1 \over \chi (id)} \sum_{\sigma \in G} \chi (\sigma) \prod^n_{i = 1} a_{i, \sigma (i)} \] is never larger than \(\text{per} (A)\). The PDC has attracted much attention in the recent years; a survey is given by \textit{R. Merris} [Current Trends in Matrix Theory, 213-223 (1987; Zbl 0671.15008)]. However, little is known about the MEC; a proof in the case \(n = 3\) was provided by \textit{R. Bapat} and \textit{V. Sunder} [Linear Algebra Appl. 76, 153-163 (1986; Zbl 0602.15008)]. In the present paper the following is proved: Should the MEC fail in the real case, then the smallest \(n\) for which it fails must be such that it fails at a singular matrix having certain properties (including zero row sums).
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 614183
zbMATH Keywords: permanents
zbMATH Keywords: positive semidefinite Hermitian matrices
zbMATH Keywords: Schur power matrix
zbMATH Keywords: maximum eigenvalue conjecture
zbMATH Keywords: largest eigenvalue
zbMATH Keywords: permanental dominance conjecture
zbMATH Keywords: matrix function
Wikidata QID: Q123348056
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)90117-1
OpenAlex ID: W2024823407
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description: scientific article; zbMATH DE number 614183