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