Page 452 - Schaum's Outline of Theory and Problems of Signals and Systems

APP. A] REVIEW OF MATRIX THEORY 439

With any such function we can associate a function of an N x N matrix A:
m
f(A) =a,I +a,A +a2A2 + . = x , ~ ~ (A.48)
a
k=O
If A is a diagonal matrix D in Eq. (A.421, then using Eq. (A.431, we have

If P diagonalizes A, that is [Eq. (A.4411,
P-'AP =A
then we have
A = PAP-'

and

Thus, we obtain
f(~) P~(A)P-
=
Replacing D by A in Eq. (A.491, we get

where A, are the eigenvalues of A.

C. The Cayley-Hamilton Theorem:
Let the characteristic polynomial c(A) of an N x N matrix A be given by [Eq. (A.3111

c(A) = IAI - AJ = AN + CN-i AN-] + . . . +c1A + C,
The Cayley-Hamilton theorem states that the matrix A satisfies its own characteristic
equation; that is,
c(A) = AN + + +c,A + cOI = 0 (A.54)