Combinatorial proofs of multivariate Cayley--Hamilton theorems

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.