On Generalized Eigenvalues of MAX Matrices to MIN Matrices and LCM Matrices to GCD Matrices

On Generalized Eigenvalues of MAX Matrices to MIN Matrices and LCM Matrices to GCD Matrices

Jorma K. Merikoski and Pentti Haukkanen
Faculty of Information Technology and Communication Sciences
FI-33014 Tampere University
Finland

Antonio Sasaki
Centre de Mathématiques Appliquées
École Nationale Supérieure des Mines de Paris
Université Paris Sciences et Lettres
FR-06560 Valbonne
France

Timo Tossavainen
Department of Health, Education and Technology
Luleå University of Technology
SE-97187 Luleå
Sweden

Abstract:

We determine, for every n ≥ 1, the generalized eigenvalues of an n × n MAX matrix to the corresponding MIN matrix. We also show that a similar result holds for the generalized eigenvalues of an n × n LCM matrix to the corresponding GCD matrix when n ≤ 4, but breaks down for n > 4. In addition, we prove Cauchy's interlacing theorem for generalized eigenvalues, and we conjecture an unexpected connection between the OEIS sequence A004754 and the appearance of –1 as a generalized eigenvalue in the LCM–GCD setting.

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Received May 5 2025; revised version received November 18 2025. Published in Journal of Integer Sequences, December 2 2025.

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