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

TL;DR

Merikoski, Haukkanen, Sasaki, and Tossavainen study generalized eigenvalues of MAX matrices to MIN matrices and of LCM matrices to GCD matrices. They derive a closed-form spectrum for the MAX--MIN pair: for any $S=\{s_1<\cdots<s_n\}$, the g-eigenvalues of ${\bf M}_S$ to ${\bf N}_S$ are $\lambda_1=\sqrt{s_n/s_1}$, $\lambda_2=\cdots=\lambda_{n-1}=-1$, $\lambda_n=-\sqrt{s_n/s_1}$, and provide a concrete $n=4$ example; for $T=\{1,\dots,n\}$ the spectrum is known for $n\le 4$ but breaks down for $n>4$, with a generalized Cauchy interlacing theorem and a conjectural link to OEIS sequence A004754 guiding larger-n behavior. They also show that reordering index sets does not affect the g-eigenvalues and discuss the challenges of computing high-degree g-eigenpolynomials, suggesting a stable test using $\det({\bf L}_T+\bf G_T)$. The work thus reveals a robust spectral structure for MAX--MIN matrices and a delicate, number-theoretically influenced pattern for LCM--GCD matrices, including a proposed binary-pattern criterion for the appearance of $-1$.

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.

Theorems & Definitions (8)

• Theorem 1
• proof
• Remark 2
• Remark 3
• Remark 4
• Conjecture 5
• Theorem 6
• proof