Combinatorial proofs of multivariate Cayley–Hamilton theorems

Abstract

We give combinatorial proofs of two multivariate Cayley–Hamilton type theorems. The first one is due to Phillips (1919) [10] involving 2k matrices, of which k commute pairwise. The second one uses 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 (2017) [3]. We prove a Phillips-type generalization of the Bapat–Roy theorem, which involves 2nk matrices, where n is the size of the matrices, among which nk commute pairwise. Our proofs generalize the univariate proof of Straubing (1983) [11] for the original Cayley–Hamilton theorem in a nontrivial way, and involve decorated permutations and decorated paths.