Let A,B be symmetric nxn real matrices with B positive definite and strictly diagonally dominant. We derive two localization sets for the complementarity eigenvalues of (A,B), the tightest one assuming additionally that A is copositive. This extends He-Liu-Shen sets to the case where B is not the identity. Moreover, we compare the computable bounds obtained from these new sets with the extreme classical generalized eigenvalues.
@article{sasaki2026localization,author={Sasaki, Antonio and Demassey, Sophie and Sessa, Valentina},title={Localization of complementarity eigenvalues},journal={Communications in Optimization Theory},year={2026},note={In press},}
2025
Published
On Generalized Eigenvalues of MAX Matrices to MIN Matrices and LCM Matrices to GCD Matrices
Jorma K. Merikoski, Pentti Haukkanen, Antonio Sasaki, and Timo Tossavainen
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 conjecture an unexpected connection between the OEIS sequence A004754 and the appearance of −1 as a generalized eigenvalue in the LCM–GCD setting.
@article{merikoski2025generalized,author={Merikoski, Jorma K. and Haukkanen, Pentti and Sasaki, Antonio and Tossavainen, Timo},title={On Generalized Eigenvalues of {MAX} Matrices to {MIN} Matrices and {LCM} Matrices to {GCD} Matrices},journal={Journal of Integer Sequences},year={2025},volume={28},number={7},}