Page 411 - Schaum's Outline of Theory and Problems of Signals and Systems

398 STATE SPACE ANALYSIS

7.25. Find An for

The characteristic polynomial c(A) of A is

Thus, the eigenvalues of A are A, = A, = 2. We use the Cayley-Hamilton theorem to evaluate
An. By Eq. (7.27) we have

where b, and b, are determined by setting A = 2 in the following equations [App. A,
Eqs. (A.59) and (A.60)I:
b, +b,A =An
b, = n~"-'

Thus,
b, + 2b, = 2"

b, =n2"-'
from which we get
b,= (1 -n)2" b, = n2"-'

and

7.26. Consider the matrix A in Prob. 7.25. Let A be decomposed as

N = [: i]
where D = [i i] and

(a) Show that N2 = 0.
(b) Show that D and N commute, that is, DN = ND.
(c) Using the results from parts (a) and (b), find An.
(a) By simple multiplication we see that

(b) Since the diagonal matrix D can be expressed as 21, we have

DN = 21N = 2N = 2NI = N(2I) = ND
that is, D and N commute.