Page 454 - Schaum's Outline of Theory and Problems of Signals and Systems

APP. A] REVIEW OF MATRIX THEORY

EXAMPLE A.14 Let

The characteristic polynomial is

and the minimal polynomial is
m(A) =A -a

since

Notes:
1. Every eigenvalue of A is a zero of m(A).
2. If all the eigenvalues of A are distinct, then c(A) = m(A).
3. C(A) is divisible by m(A).
4. m(A) may be used in the same way as c(A) for the expression of higher powers of A in
terms of a limited number of powers of A.

It can be shown that m(A) can be determined by

where d(A) is the greatest common divisor (gcd) of all elements of adj(A1 -A).

EXAMPLE A.15 Let

Then