CayleyHamilton

The Cayley-Hamilton theorem is a fundamental result in linear algebra concerning square matrices. It states that every square matrix over a commutative ring satisfies its own characteristic polynomial. More formally, if A is an n x n matrix with entries in a commutative ring R, and p(lambda) is its characteristic polynomial, then p(A) = 0, where 0 is the n x n zero matrix. The characteristic polynomial of A is defined as det(A - lambda I), where det denotes the determinant and I is the identity matrix.

This theorem has significant theoretical and practical implications. Theoretically, it provides a direct relationship between a matrix and its characteristic polynomial, which encodes important information about the matrix, such as its eigenvalues. Practically, it can be used to compute powers of a matrix. If p(lambda) = c_n lambda^n + c_{n-1} lambda^{n-1} + ... + c_1 lambda + c_0, then p(A) = c_n A^n + c_{n-1} A^{n-1} + ... + c_1 A + c_0 I = 0. This equation can be rearranged to express A^n as a linear combination of lower powers of A (A^{n-1}, ..., A, I), which simplifies matrix exponentiation and solving systems of linear differential equations. The Cayley-Hamilton theorem holds for matrices over any commutative ring, not just fields like real or complex numbers.