On generalized eigenvalues of MAX matrices to MIN matrices and of LCM matrices to GCD matrices

TL;DR

This paper investigates the generalized eigenvalues of MAX and MIN matrices, as well as LCM and GCD matrices, establishing results for small sizes and revealing interesting conjectures and interlacing properties.

Contribution

It provides explicit results for generalized eigenvalues of MAX-MIN matrices and LCM-GCD matrices for small sizes, and proves a generalized interlacing theorem.

Findings

Explicit eigenvalues for MAX-MIN matrices for all n

Results for LCM-GCD matrices hold for n ≤ 4 but not beyond

Proves Cauchy's interlacing theorem for generalized eigenvalues

Abstract

We determine, for any n $\ge$ 1, the generalized eigenvalues of an n x n MAX matrix to the corresponding MIN matrix. We also show that a similar result holds for the generalized eigenvalues of an nxn LCM matrix to the corresponding GCD matrix when n $\le$ 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.