Page 424 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 71 STATE SPACE ANALYSIS

7.41. Repeat Prob. 7.39 using the spectral decomposition method.
Since all eigenvalues of A are distinct, by Eq. (7.33) we have
1 3
El = - (A-A21)=A+31= [-6 -:]
AI -A2
1
-:I
E2= - (A-AII) = -(A+21) = [-:
A2 - A1
Then by Eq. (7.70) we obtain

7.42. Repeat Prob. 7.39 using the Laplace transform method.

First, we must find (SI -A)-'

Then, by Eq. (7.71) we obtain

Again we note that when the eigenvalues of A are all distinct, the spectral decomposition
method is computationally the most efficient method of evaluating eA'.

7.43. Find eA' for

The characteristic polynomial c(A) of A is

=h2+4~+3=(~+ 1)(A +3)
Thus, the eigenvalues of A are A, = - 1 and A, = -3. Since all eigenvalues of A are distinct, by