Applications of the Cayley-Hamilton Theorem

In linear algebra, the Cayley–Hamilton theorem states that every square matrix satisfies its own characteristic polynomial.

Suppose A is a given n × n matrix and I is the n × n identity matrix. In that case, the characteristic polynomial of A is defined as

f_A(x) = |xI – A|, the determinant of xI – A, where x is a variable. f_A(x) is a polynomial in x of degree n with the leading coefficient 1. So f_A(x) = x^n + a_{n-1}x^{n-1}+ … + a_1 x +a_0. and we call f_A(x) the characteristic polynomial of A. Then the Cayley-Hamilton theorem says that f_A(A) = 0, namely, A^n + a_{n-1}A^{n-1} + … + a_1A + a_0I = 0. We will check this result with an example. In this short course, we will give two applications of this result.

The first application is to find the inverse of an invertible matrix A.  We first note that a_0 = (-1)^n |A|. So A is invertible if and only if a_0 is non-zero, and in this case, A ^{-1} = – 1/a_0 (A^{n-1} + a_{n-1}A ^{n-2} + … + a_1 I ). We will give an example to illustrate this result.

The second application is to find the powers of the square matrix A. We will assume that A has n distinct eigenvalues. We first give a short discussion of the Vandernonde matrix associated with the n eigenvalues to conclude that it is invertible in the current case, and then reduce the problem to solving a system of linear equations with the coefficient matrix to be the Vandermonde matrix associated with the n eigenvalues. We will also give an example to illustrate the result.