Page 412 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 71 STATE SPACE ANALYSIS

(c) Using the binomial expansion and the result from part (b), we can write

Since N* = 0, then Nk = 0 for k 2 2, and we have

Thus [see App. A, Eq. (A.431,

which is the same result obtained in Prob..7.25.
Note that a square matrix N is called nilpotent of index r if Nr-' # 0 and Nr = 0.

7.27. The minimal polynomial m(A) of A is the polynomial of lowest order having 1 as its
leading coefficient such that m(A) = 0. Consider the matrix
0 0

(a) Find the minimal polynomial m(A) of A.
(b) Using the result from part (a), find An.
(a) The characteristic polynomial c(A) of A is

Thus, the eigenvalues of A are A, = - 3 and A, = A, = 2. Consider

Now

Thus, the minimal polynomial of A is

(b) From the result from part (a) we see that An can be expressed as a linear combination of
I and A only, even though the order of A is 3. Thus, similar to the result from the