Ordinary Differential Equations and Dynamical Systems Vol2 · Problem 2.4

Existence Theory

Mathematics — Calculus · cayley-hamilton-linear-systems

Problem

Let

A

be an

n \times n

matrix over

\mathbb{R}

with

p(\lambda) = \lambda^n + c_{n-1}\lambda^{n-1} + \cdots + c_1\lambda + c_0

as its characteristic polynomial. The Cayley-Hamilton Theorem states that

p(A) = 0

. Use this fact to prove that the components

x_j(t)

of the solution

x(t)

to the linear system

x' = Ax

satisfy the

n

th order equation

p(D)u = u^{(n)} + c_{n-1}u^{(n-1)} + \cdots + c_1 u' + c_0 u = 0.