Page 136 - Numerical methods for chemical engineering

122 3 Matrix eigenvalue analysis

For a matrix to be unitary, its column vectors must be orthogonal, as
 
[1] H
— w —  | | 
H  .  [1]
W W =  . .   w ... w [N] 
 
[N] H | |
— w —
 [1] [1] [1] [N] 
w · w ... w · w
. .
. .  = I (3.117)
 
. .
= 
[N] [1] [N] [N]
w · w ... w · w
Therefore, it is always possible, for any normal matrix A, to find a complete, orthonormal
basis for C N whose members are eigenvectors of A. One can write any vector v ∈ C N as
the spectral decomposition
v = c 1 w [1] + c 2 w [2] + ··· + c N w [N] w [ j] ∈ C N (3.118)
Aw [ j] = λ j w [ j] w [ j] · w [k] = δ jk c j = w [ j] · v QED
H
CorollaryENV1-1 Let us write a normal matrix A in Jordan normal form as A = W W .
H
H
H
Taking the Hermitian conjugate, A = W W . Therefore, the normal matrices A and A H
H
share the same set of eigenvectors, and the eigenvalues of A are the complex conjugates
of those of A,
H
Aw [ j] = λ j w [ j] ⇒ A w [ j] = λ j w [ j] (3.119)
H
Theorem EVN2 If A is Hermitian, A = A , all of its eigenvalues are real.
H
Proof If A = A , then by writing the matrices in normal form,
H
H H
H
A = W W H A = (W W ) = W W H (3.120)
we see the diagonal matrix of eigenvalues must also be Hermitian,
= H (3.121)

Thus for each j = 1, 2,..., N,λ j = λ j , and every eigenvalue must be real. QED
N
Corollary ENV2-1 (spectral decomposition of ) Let A be a real, symmetric matrix,
[ j]
T
A = A, and thus Hermitian. From Aw [ j] = λ j w , as both A and λ j are real, it is always
possible to find a real set of mutually orthonormal eigenvectors for A. Therefore, we may
N
write any vector v ∈ as the eigenvector expansion
v = c 1 w [1] + c 2 w [2] +· · · + c N w [N] w [ j] ∈ N
(3.122)
Aw [ j] = λ j w [ j] w [ j] · w [k] = δ jk c j = w [ j] · v ∈
T
Definition A real symmetric matrix A is said to be positive-definite if v · Av = v Av > 0
N
for all v ∈ . Let the eigenvalues and orthonormal eigenvectors of A satisfy Aw [ j] =
N
[ j]
N
λ j w , w [ j] · w [k] = δ jk , w [ j] ∈ ,λ j ∈ . Thus, we can write any v ∈ as the linear
combination
N N
[ j]
[ j]
[ j]

v = w · v w = c j w (3.123)
j=1 j=1