On eigenvalues of meet and join matrices associated with incidence functions

✍ Scribed by Pauliina Ilmonen; Pentti Haukkanen; Jorma K. Merikoski

• Publisher: Elsevier Science
• Year: 2008
• Tongue: English
• Weight: 166 KB
• Volume: 429
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2008.04.014

✦ Synopsis

Let (P , , ∧) be a locally finite meet semilattice. Let S = {x 1 , x 2 , . . . , x n }, x i x j ⇒ i j, be a finite subset of P and let f be a complex-valued function on P . Then the n × n matrix (S) f , where

is called the meet matrix on S with respect to f . The join matrix on S with respect to f is defined dually on a locally finite join semilattice.

In this paper, we give lower bounds for the smallest eigenvalues of certain positive definite meet matrices with respect to f on any set S. We also estimate eigenvalues of meet matrices respect to any f on meet closed set S and with respect to semimultiplicative f on join closed set S. The same is carried out dually for join matrices.