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In-Depth Information
S
ij
:¼ δ
ij
s
j
,
ð
8
:
12
Þ
where
λ
p
,
j ¼
1,
s
j
:¼
...
,
n
are the
singular values
of
A
. We have thus derived the well-known
singular value
decomposition
(SVD).
mn
. Then there is a unique
Proposition 8.2 (cf. Lemma 7.3.1 in [HJ85])
Let A
∈R
sequence s
1
... s
m
such that
A ¼ USV
T
,
ð
8
:
13
Þ
nn
.
The values s
j
are referred to as
singular values
of A, the columns of U as
left
, and
those of V as
right singular vectors of
A.
mm
,V
where S is as defined in (
8.12
), for some unitary matrices U
∈ R
∈ R
Example 8.3
For our Example 8.1 of a web shop with matrix
0
@
1
A
,
01 05
1511
0501
A ¼
we approximately obtain the following SVD:
0
1
0
1
11
:
49
0
0
0
10
:
30
:
1
@
A
,
U ¼
@
A
,
S ¼
06
:
88
0
0
0
:
2
0
:
7
0
:
7
0
0
0
:
77
0
0
:
1
0
:
70
:
7
0
1
0
:
02
0
:
1
0
:
91
0
:
39
@
A
:
0
:
24
0
:
96
0
:
07
0
:
09
V ¼
0
:
84
0
:
24
0
:
02
0
:
06
0
:
45
0
:
02
0
:
41
0
:
94
■
Labeling the left and right singular vectors as
U ¼
:[
u
1
,
...
,
u
m
],
V ¼
:[
v
1
,
...
,
v
n
],
we obtain a polyadic representation
A ¼
X
r
s
j
u
j
v
j
,
j¼
1
where
r
0
.
It may easily be verified that
r ¼ rank A
, i.e., equal the
minimum number terms in a polyadic representation of
A
or, equivalently, the
dimension of the range of
A.
It is thus also obvious that {
u
1
,
:¼
max
k
s
k
>
...
,
u
r
}isan
