Page 453 - Schaum's Outline of Theory and Problems of Signals and Systems

440 REVIEW OF MATRIX THEORY [APP. A

EXAMPLE A.13 Let

Then, its characteristic polynomial is

and

Rewriting Eq. (A.54), we have

Multiplying through by A and then substituting the expression (A.55) for AN on the right and
rearranging, we get

By continuing this process, we can express any positive integral power of A as a linear combination of
I,A,. ..,A~-'. Thus, f(A) defined by Eq. (A.48) can be represented by

In a similar manner, if A is an eigenvalue of A, then f(A) can also be expressed as
N- l
f(A) =b,+ blA + .a. +b,-,~~-l C b,Arn ( A.58)
=
m =O
Thus, if all eigenvalues of A are distinct, the coefficients bm (rn = 0,1,. . . , N - 1) can be determined
by the following N equations:

If all eigenvalues of A are not distinct, then Eq. (A.59) will not yield N equations. Assume that an
eigenvalue A, has multiplicity r and all other eigenvalues are distinct. In this case differentiating both
sides of Eq. (A.58) r times with respect to A and setting A = A;, we obtain r equations corresponding
to Ai:

Combining Eqs. (A.59) and (A.601, we can determine all coefficients bm in Eq. (A.57).

D. Minimal Polynomial of A:

The minimal (or minimum) polynomial m(h) of an N x N matrix A is the polynomial
of lowest degree having 1 as its leading coefficient such that m(A) = 0. Since A satisfies its
characteristic equation, the degree of m(A) is not greater than N.