Graphics Reference
In-Depth Information
DV
t
V
t
MU
=
M
=
U
D
=
=
(a)
(b)
Figure 10.8: (a) An n
×
k matrix, with n
>
k, factors as a product of an n
×
n matrix with
orthonormal columns (indicated by the vertical stripes on the first rectangle), a diagonal
k
×
k matrix, and a k
×
k matrix with orthonormal rows (indicated by the horizontal stripes),
which we write as
UDV
T
,where
U
and
V
have orthonormal
columns.
(b) An n
kmatrix
with n
<
k is written as a similar product; note that the diagonal matrix in both cases is
square, and its size is the smaller of n and k.
×
guideline, if the ratio of the largest to the smallest singular values is very large
(say, 10
6
), then numerical computations with the matrix are likely to be unstable.
Inline Exercise 10.16:
The singular value decomposition is not unique. If we
negate the first row of
V
T
and the first column of
U
in the SVD of a matrix
M
,
show that the result is still an SVD for
M
.
In the special case where
n
=
k
(the one we most often encounter), the matri-
ces
U
and
V
are both square and represent change-of-coordinate transformations
in the domain and codomain. Thus, we can see the transformation
T
(
x
)=
Mx
(10.41)
as a sequence of three steps: (1) Multiplication by
V
T
converts
x
to
v
-coordinates;
(2) multiplication by
D
amounts to a possibly nonuniform scaling along each
axis; and (3) multiplication by
U
treats the resultant entries as coordinates in the
u
-coordinate system, which then are transformed back to standard coordinates.
Howdowefind
U
,
D
, and
V
? In general it's relatively difficult, and we rely on
numerical linear algebra packages to do it for us. Furthermore, the results are by no
means unique: A single matrix may have multiple singular value decompositions.
For instance, if
S
is
any n
×
n
matrix with orthonormal columns, then
I
=
SIS
T
(10.42)
is one possible singular value decomposition of the identity matrix. Even though
there are many possible SVDs, the singular values are the same for all decompo-
sitions.
The
rank
of the matrix
M
, which is defined as the number of linearly inde-
pendent columns, turns out to be exactly the number of nonzero singular values.
Again, in the special case where
n
=
k
so that
U
and
V
are square, it's easy to
compute
M
−
1
if you know the SVD:
M
−
1
=
V
D
−
1
U
T
,
(10.43)
