Combinatorial proofs of multivariate Cayley--Hamilton theorems

TL;DR

This paper provides combinatorial proofs for multivariate Cayley--Hamilton theorems, including generalizations involving multiple matrices and the mixed discriminant, using decorated permutations and paths.

Contribution

It introduces new combinatorial proofs for multivariate Cayley--Hamilton theorems, extending previous results to more matrices and commuting conditions.

Findings

Proved a Phillips-type generalization involving 2nk matrices.

Extended the Cayley--Hamilton theorem for mixed discriminants.

Utilized decorated permutations and paths in proofs.

Abstract

We give combinatorial proofs of two multivariate Cayley--Hamilton type theorems. The first one is due to Phillips (Amer. J. Math., 1919) involving $2k$ matrices, of which $k$ commute pairwise. The second one regards the mixed discriminant, a matrix function which has generated a lot of interest in recent times. Recently, the Cayley--Hamilton theorem for mixed discriminants was proved by Bapat and Roy (Comb. Math. and Comb. Comp., 2017). We prove a Phillips-type generalization of the Bapat--Roy theorem involving $2nk$ matrices, where $n$ is the size of the matrices, among which $nk$ commute pairwise. Our proofs generalize the univariate proof of Straubing (Disc. Math., 1983) for the original Cayley--Hamilton theorem in a nontrivial way, and involve decorated permutations and decorated paths.