CS205 – Class 6
Reading:
Heath 3.6 (p137143), 4.7 (p202)
Singular Value Decomposition (SVD) contd.
1. SVD is a transformation into a diagonal axis aligned space.
a. Transform
b
into the space spanned by
T
U
,
b
b
U
x
V
x
V
U
U
T
T
T
T
ˆ
. No information is lost
going from
b
to
b
ˆ
because
T
U
is square and orthogonal.
b. Replace
x
V
T
by
x
ˆ to get a diagonal system,
b
x
x
V
T
ˆ
ˆ
.
c. Now solve the system
b
x
ˆ
ˆ
simply by scaling elements of
b
ˆ
by the singular values.
d. The original x is then recovered as
x
V
x
ˆ
.
e. Essentially the SVD solves the matrix by transforming the vectors in a space with eigenvectors along
the unit axis.
2.
T
T
i i i
i
A U
V
u v
proof: define
)
,
min(
n
m
l
,
U
ˆ
the first
l
columns of
U
,
ˆ
the square
l
l
submatrix from the upper left
corner of
,
V
ˆ
the first
l
columns of
V
. Then
T
l
l
T
l
T
l
T
l
l
T
T
v
v
u
u
v
v
u
u
V
U
V
U
A
1
1
1
1
1
1
ˆ
ˆ
ˆ
l
i
T
i
i
i
l
i
n
i
m
i
i
i
m
i
i
n
i
i
i
i
i
i
l
i
n
i
m
i
i
l
i
i
m
i
i
l
i
n
i
i
i
l
i
i
i
i
v
u
v
u
v
u
v
u
v
u
v
u
v
u
v
u
v
u
1
1
1
1
1
1
1
1
1
1
1
1
1
1
a. Note that “zero” or “small”
i
produce terms that contribute little to the sum, and that large
i
produce terms that contribute significantly to the sum.
b. If the “zero” or “small”
i
are omitted from the summation, one obtains a matrix with lower rank. For
example, if only the first k terms are summed, the result has rank k.
i. Moreover, it can be shown that this new rank k matrix is the closest rank k matrix to A in both
the L
2
and the Frobenius norm.
ii. This is the key idea in PCA, clustering/data mining algorithms, etc.
3. The “pseudoinverse” of a matrix A is defined by
T
A
V
U
where
is obtained from
by replacing
all “nonzero”
i
with 1/
i
, and leaving all the zero entries
identically zero
.
a. If A is square and nonsingular (
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 Spring '07
 Numerical Analysis, A. Newton, xk

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