Proof of Cayley-Hamilton theorem using polynomials over the algebra of module endomorphisms

A Proof of the Cayley-Hamilton Theorem

Let M (n, n) be the set of all n × n matrices over a commutative ring with identity. Then the Cayley Hamilton Theorem states: Theorem. Let A ∈ M (n, n) with characteristic polynomial det(tI − A) = c 0 t n + c 1 t n−1 + c 2 t n−2 + · · · + c n. Then c 0 A n + c 1 A n−1 + c 2 A n−2 + · · · + c n I = 0. In this note we give a variation on a standard proof (see [1], for example). The idea is to use...

On a Diagrammatic Proof of the Cayley-hamilton Theorem

This note concerns a one-line diagrammatic proof of the CayleyHamilton Theorem. We discuss the proof’s implications regarding the “core truth” of the theorem, and provide a generalization. We review the notation of trace diagrams and exhibit explicit diagrammatic descriptions of the coefficients of the characteristic polynomial, which occur as the n + 1 “simplest” trace diagrams. We close with ...

The Cayley-Hamilton theorem

Theorem 1. Let A be a n × n matrix, and let p(λ) = det(λI − A) be the characteristic polynomial of A. Then p(A) = 0.

The Cayley-Hamilton Theorem

This document contains a proof of the Cayley-Hamilton theorem based on the development of matrices in HOL/Multivariate Analysis.

The Cayley–Hamilton Theorem