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Thus, the same algorithm that computes the eigenpairs for
M
T
M
gives us the matrix
V
for
the SVD of
M
itself. It also gives us the singular values for this SVD; just take the square
roots of the eigenvalues for
M
T
M
.
Only
U
remains to be computed, but it can be found in the same way we found
V
. Start
with
MM
T
=
U
Σ
V
T
(
U
Σ
V
T
)
T
=
U
Σ
V
T
V
Σ
U
T
=
U
Σ
2
U
T
Then by a series of manipulations analogous to the above, we learn that
MM
T
U
=
U
Σ
2
That is,
U
is the matrix of eigenvectors of
MM
T
.
A small detail needs to be explained concerning
U
and
V
. Each of these matrices have
r
columns, while
M
T
M
is an
n
×
n
matrix and
MM
T
is an
m
×
m
matrix. Both
n
and
m
are at
least as large as
r
. Thus,
M
T
M
and
MM
T
should have an additional
n
−
r
and
m
−
r
eigen-
pairs, respectively, and these pairs do not show up in
U
,
V
, and Σ. Since the rank of
M
is
r
,
all other eigenvalues will be 0, and these are not useful.
11.3.7
Exercises for Section 11.3
EXERCISE
11.3.1 In
Fig. 11.11
is a matrix
M
. It has rank 2, as you can see by observing that
the first column plus the third column minus twice the second column equals
0
.
Figure 11.11
Matrix
M
for
Exercise 11.3.1
(a) Compute the matrices
M
T
M
and
MM
T
.
!
(b) Find the eigenvalues for your matrices of part (a).
(c) Find the eigenvectors for the matrices of part (a).
(d) Find the SVD for the original matrix
M
from parts (b) and (c). Note that there are
only two nonzero eigenvalues, so your matrix Σ should have only two singular val-
ues, while
U
and
V
have only two columns.
(e) Set your smaller singular value to 0 and compute the one-dimensional approxima-
tion to the matrix
M
from
Fig. 11.11
.
