The solution of the Loewy–Radwan conjecture

Omladič, Matjaž (author), Šivic, Klemen (author)

A seminal result of Gerstenhaber gives the maximal dimension of a linear space of nilpotent matrices. It also exhibits the structure of such a space when the maximal dimension is attained. Extensions of this result in the direction of linear spaces of matrices with a bounded number of eigenvalues have been studied. In this paper, we answer what is perhaps the most general problem of the kind as proposed by Loewy and Radwan, by solving their conjecture in the positive. We give the maximal dimension of a vector space of $n \times n$ matrices with no more than $k < n$ eigenvalues. We also exhibit the structure of the spaces for which this dimension is attained.

• Language: English
• Keywords: linear space of matrices, eigenvalues, maximal number of distinct eigenvalues, dimension, structure, representation of groups
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2024
• Number of pages: Str. 2592–2618
• Numbering: Vol. 72, no. 15
• PID: 20.500.12556/RUL-163046
• UDC: 512
• ISSN on article: 0308-1087
• DOI: 10.1080/03081087.2023.2277207
• COBISS.SI-ID: 178861571
• Publication date in RUL: 01.10.2024

Record is a part of a journal

• Title: Linear and multilinear algebra
• Shortened title: Linear multilinear algebra
• Publisher: Taylor & Francis
• ISSN: 0308-1087
• COBISS.SI-ID: 25872128

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