Page 134 - Numerical methods for chemical engineering

120 3 Matrix eigenvalue analysis

N
orthonormal basis for C . Then, the matrix
 H 
— w —
  H
| | |  — u [2] 


U [N] =  w u [2] ... u [N]  U [N] H =  . —  (3.101)

 . . 
| | |  
[N] H
— u —
) U
) U
is unitary, (U [N] H [N] = I, as{(U [N] H [N] } mn = u [m] ·u [n] = δ mn . Then,
   
| | | | | |
AU [N] = A  w u [2] ... u [N]  =  λw Au [2] ... Au [N]  (3.102)
| | | | | |
and
 H 
— w —
[2] H  | | | 
 — u 

[N] H [N] —  [2] [N]

U AU =  .   λw Au ... Au 
 . . 
  | | |
[N] H
— u —
 H 
λ — c —
| ...
 b 22 b 2N 
. . 
  (3.103)
= 
 0 . . . . 
| b N2 ... b NN
[n]
b mn = u [m] · Au . That is, in terms of an (N − 1) × (N − 1) matrix B,
H

[N] H [N] λ c
U AU = (3.104)
0 B
H
H
If the theorem holds for B, we can write V BV = T , for V = V −1 and T upper triangular.
Then, defining the N × N unitary matrix
T T
1 0 [N−1] H 1 0
[N−1]

U = U = H (3.105)
0 V 0 V
we have
T H T

[N−1] H [N] H [N] [N−1] 1 0 λ c 1 0

U U AU U =
0 V H 0 B 0 V
H
λ c
= H (3.106)
0 (V BV )
H
As U = U [N] U [N−1] is unitary and V BV = T, then
H
λ c
H
U AU = R R = (3.107)
0 T
Thus, if the theorem holds for (N − 1) × (N − 1) matrices, it holds for N × N matrices as
well. QED