Page 156 - Numerical methods for chemical engineering

142 3 Matrix eigenvalue analysis

Let us assume that our matrix has more rows than columns, M ≥ N. We then can write A
in the form of a Singular Value Decomposition (SVD)
A = W V T (3.232)

is a M × N diagonal matrix containing the singular values, W is an orthogonal M × M
matrix whose columns are the M left singular vectors of A, and V is an orthogonal N × N
matrix whose columns are the N right singular vectors of A.

 
σ 1
 
σ 2
 
.
 
 . .   | | | 
 
  [1] [2]
W =  w w ... w [M] 
=  σ N 
 
| | |
 0 
  T −1
.
 .  W = W
 . 
0
 
| | |
 [1] [2]
V = v v ... v [N]  (3.233)
| | |
For M < N, the SVD also exists, but now is
 
σ 1
σ 2
 
=   (3.234)
...
 
000
σ M
T
T
T
T
T
T
T
and σ M+1 =· · · = σ N = 0. As A A = (V W )(W V ) = V (W W) V , we note
T
that as W is orthogonal, W W = I M , I M being the M × M identity matrix, and thus we
T
obtain the Jordan normal form for A A
T
T
A A = V ( )V T (3.235)
T
T
Therefore, = is a diagonal matrix containing the eigenvalues of A A, and the right
T
singular vectors of A are the eigenvectors of A A,
2 [ j]
T
A Av [ j] = σ v (3.236)
j
Previously,wehavedefinedtherankofasquarematrixasthenumberoflinearly-independent
columns (or rows); however, we can now extend this definition and provide a means for its
calculation.
Definition The rank of an M × N matrix A is the number of its nonzero singular values.
Above we have treated the case of a real matrix A. For a complex matrix A, the SVD is
H
A = U V , where now U and V are unitary. contains the nonnegative singular values,
as is the case when A is real.