Randomized methods for computing joint eigenvalues, with applications to multiparameter eigenvalue problems and root finding

He, Haoze (author), Kressner, Daniel (author), Plestenjak, Bor (author)

It is well known that a family of $n \times n$ commuting matrices can be simultaneously triangularized by a unitary similarity transformation. The diagonal entries of the triangular matrices define the $n$ joint eigenvalues of the family. In this work, we consider the task of numerically computing approximations to such joint eigenvalues for a family of (nearly) commuting matrices. This task arises, for example, in solvers for multiparameter eigenvalue problems and systems of multivariate polynomials, which are our main motivations. We propose and analyze a simple approach that computes eigenvalues as one-sided or two-sided Rayleigh quotients from eigenvectors of a random linear combination of the matrices in the family. We provide some analysis and numerous numerical examples, showing that such randomized approaches can compute semisimple joint eigenvalues accurately and lead to improved performance of existing solvers.

• Language: English
• Keywords: commuting matrices, joint eigenvalues, Rayleigh quotient, randomized numerical linear algebra, multiparameter eigenvalue problems, polynomial systems
• 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: 2025
• Number of pages: Str. 861–892
• Numbering: Vol. 100, iss. 3
• PID: 20.500.12556/RUL-175108
• UDC: 519.6
• ISSN on article: 1017-1398
• DOI: 10.1007/s11075-024-01971-0
• COBISS.SI-ID: 215046403
• Publication date in RUL: 16.10.2025

Record is a part of a journal

• Title: Numerical algorithms
• Shortened title: Numer. algorithms
• Publisher: Springer Nature
• ISSN: 1017-1398
• COBISS.SI-ID: 30694400

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: komutirajoče matrike, skupne lastne vrednosti, Rayleighov kvocient, verjetnostna numerična linearna algebra, večparametrični problemi lastnih vrednosti, polinomski sistemi