On meet hypermatrices and their eigenvalues

Abstract

Let be a locally finite meet semilattice. Let

be a finite subset of , and let be a complex-valued function on . The -dimensional hypermatrix of order , , given by

is called the order meet hypermatrix on with respect to . We consider -dimensional meet hypermatrices of order . As an example, we consider GCD hypermatrices. We examine the structure of order meet hypermatrices with respect to , and provide a structure theoretical result that is a generalization of a known result for meet matrices. We also give a region in which all the eigenvalues of an -dimensional order meet hypermatrix with respect to a real-valued lie, and using that we obtain results concerning positive definiteness and E-eigenvalues of meet hypermatrices. Characteristics of meet matrices and the eigenvalues of supersymmetric hypermatrices are under active research, but the eigenvalues of GCD and related hypermatrices have not hitherto been considered in the literature.

Keywords:

• meet hypermatrix
• GCD matrix
• semilattice
• eigenvalue
• E-eigenvalue

AMS Subject Classifications:

• 11C20
• 06A12
• 15A18
• 15A69

Acknowledgements

The author wishes to thank the anonymous referee for his/her comments and suggestions that helped to improve the paper.

Notes

No potential conflict of interest was reported by the authors.