ePrints@IIScePrints@IISc Home | About | Browse | Latest Additions | Advanced Search | Contact | Help

Combinatorial proofs of multivariate Cayley–Hamilton theorems

Ayyer, A and Sundaravaradan, N (2023) Combinatorial proofs of multivariate Cayley–Hamilton theorems. In: Linear Algebra and Its Applications, 661 . pp. 247-269.

lin_alg_app_661_247-269_2023.pdf - Published Version

Download (424kB) | Preview
Official URL: https://doi.org/10.1016/j.laa.2022.12.014


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.

Item Type: Journal Article
Publication: Linear Algebra and Its Applications
Publisher: Elsevier Inc.
Additional Information: The copyright for this article belongs to the Author(s).
Keywords: Cayley-Hamilton; Cayley-hamilton theorem; Combinatorial proof; Generalisation; matrix; Matrix functions; Mixed discriminant; Phillip' theorem; Phillips; Univariate, Graph Databases
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 09 Feb 2023 05:23
Last Modified: 09 Feb 2023 05:23
URI: https://eprints.iisc.ac.in/id/eprint/80102

Actions (login required)

View Item View Item