Page 449 - Schaum's Outline of Theory and Problems of Signals and Systems

436 REVIEW OF MATRIX THEORY [APP. A

THEOREM A.l:

Let A, (k = 1,2,. . . , i) be the distinct eigenvalues of A and let x, be the eigenvectors
associated with the eigenvalues A,. Then the set of eigenvectors x,, x,, . . . , x, are linearly
independent.
Proof. The proof is by contradiction. Suppose that x,, x,, . . . ,xi are linearly dependent.
Then there exists a,, a,, . . . , ai not all zero such that
1
a1x, + a2x2 + a . +aixi = C akxk = 0 ( A.33)
K= l
Assuming a, # 0, then by Eq. (A.33) we have

(A,I - A)(A,I - A) (A.34)

Now by Eq. (A.28)

(A,I - A)x, = (Aj - A,)x, j # k
and (A,I - A)x, = 0

Then Eq. (A.34) can be written as
a,(A2 - A,)(A, -A,) - - - (Ai - A,)x, = 0 (A.35)
Since A, (k = 1,2,. . . , i) are distinct, Eq. (A.35) implies that a, = 0, which is a contradic-
tion. Thus, the set of eigenvectors x,, x,, . . . , xi are linearly independent.

A.6 DIAGONALIZATION AND SIMILARITY TRANSFORMATION
A. Diagonalization:
Suppose that all eigenvalues of an N X N matrix A are distinct. Let x1,x2,. . . ,xN be
eigenvectors associated with the eigenvalues A,, A,, . . . , AN. Let
P= [x, x, - e m x,] (A.36)

Then AP=A[X, x, x,]
'[AX] AX2 '" AXN]
= [A,x, A2x2 ANxN]

where