The singular value decomposition is another name for the
spectral representation of a rectangular matrix. Of course if
AA is
mm-by-nn
and
m≠n
m
n
then it does not make sense to speak of the
eigenvalues of AA. We
may, however, rely on the previous section to give us relevant
spectral representations of the two symmetric matrices
That these two matrices together may indeed tell us 'everything'
about
AA can be
gleaned from
𝒩ATA=𝒩A
𝒩
A
A
𝒩
A
𝒩AAT=𝒩AT
𝒩
A
A
𝒩
A
ℛATA=ℛAT
ℛ
A
A
ℛ
A
ℛAAT=ℛA
ℛ
A
A
ℛ
A
You have proven the first of these in a previous exercise. The
proof of the second is identical. The row and column space
results follow from the first two via orthogonality.
On the spectral side, we shall now see that the eigenvalues of
AAT
A
A
and
ATA
A
A
are nonnegative and that their nonzero eigenvalues
coincide. Let us first confirm this on the adjacency matrix associated
with the unstable swing (see figure in another module)
A=0100-10100001
A
0
1
0
0
-1
0
1
0
0
0
0
1
(1)
The respective products are
AAT=100020001
A
A
1
0
0
0
2
0
0
0
1
ATA=10-100100-10100001
A
A
1
0
-1
0
0
1
0
0
-1
0
1
0
0
0
0
1
Analysis of the first is particularly simple. Its null space is
clearly just the zero vector while
λ1=2
λ1
2
and
λ2=1
λ2
1
are its eigenvalues. Their geometric multiplicities are
n1=1
n1
1
and
n2=2
n2
2
.
In
ATA
A
A
we recognize the
SS matrix from
the exercise in
another moduleand recall that its eigenvalues are
λ1=2
λ1
2
,
λ2=1
λ2
1
, and
λ3=0
λ3
0
with multiplicities
n1=1
n1
1
,
n2=2
n2
2
, and
n3=1
n3
1
. Hence, at least for this
AA, the eigenvalues of
AAT
A
A
and
ATA
A
A
are nonnegative and their nonzero eigenvalues
coincide. In addition, the geometric multiplicities of the
nonzero eigenvalues sum to 3, the rank of
AA.
The eigenvalues of
AAT
A
A
and
ATA
A
A
are nonnegative. Their nonzero eigenvalues, including
geometric multiplicities, coincide. The geometric
multiplicities of the nonzero eigenvalues sum to the rank of
AA.
If
ATAx=λx
A
A
x
λ
x
then
xTATAx=λxTx
x
A
A
x
λ
x
x
, i.e.,
∥Ax∥2=λ∥x∥2
A
x
2
λ
x
2
and so λ ≥ 0. A similar argument works
for
AAT
A
A
.
Now suppose that
λj>0
λj
0
and that
x
j
,
k
k
=
1
nj
x
j
,
k
k
=
1
nj
constitutes an orthogonal basis for the eigenspace
ℛPj
ℛ
Pj
. Starting from
ATAx
j
,
k
=λjx
j
,
k
A
A
x
j
,
k
λj
x
j
,
k
(2)
we find, on multiplying through (from the left) by
AA that
AATAx
j
,
k
=λjAx
j
,
k
A
A
A
x
j
,
k
λj
A
x
j
,
k
i.e.,
λjλj
is an eigenvalue of
AAT
A
A
with eigenvector
Ax
j
,
k
A
x
j
,
k
, so long as
Ax
j
,
k
≠0
A
x
j
,
k
0
. It follows from the first paragraph of this proof
that
∥Ax
j
,
k
∥=λj
A
x
j
,
k
λj
, which, by hypothesis, is nonzero. Hence,
∀,1≤k≤nj:y
j
,
k
≡Ax
j
,
k
λj
1
k
nj
y
j
,
k
A
x
j
,
k
λj
(3)
is a collection of unit eigenvectors of
AAT
A
A
associated with
λjλj.
Let us now show that these vectors are orthonormal for fixed
jj.
y
j
,
i
T
y
j
,
k
=1λj
x
j
,
i
T
ATA
x
j
,
k
=
x
j
,
i
T
x
j
,
k
=0
y
j
,
i
T
y
j
,
k
1
λj
x
j
,
i
T
A
A
x
j
,
k
x
j
,
i
T
x
j
,
k
0
We have now demonstrated that if
λj>0
λj
0
is an eigenvalue of
ATA
A
A
of geometric multiplicity
njnj. Reversing
the argument, i.e., generating eigenvectors of
ATA
A
A
from those of
AAT
A
A
we find that the geometric multiplicities must indeed
coincide.
Regarding the rank statement, we discern from Equation 2 that if
λj>0
λj
0
then
x
j
,
k
∈ℛATA
x
j
,
k
ℛ
A
A
. The union of these vectors indeed constitutes a basis
for
ℛATA
ℛ
A
A
, for anything orthogonal to each of these
x
j
,
k
x
j
,
k
necessarily lies in the eigenspace corresponding to a
zero eigenvalue, i.e., in
𝒩ATA
𝒩
A
A
. As
ℛATA=ℛAT
ℛ
A
A
ℛ
A
it follows that
dimℛATA=r
dimℛ
A
A
r
and hence the
njnj,
for
λj>0
λj
0
, sum to r.
Let us now gather together some of the separate pieces of the
proof. For starters, we order the eigenvalues of
ATA
A
A
from high to low,
λ1>λ2>…>λh
λ1
λ2
…
λh
and write
ATA=XΛnXT
A
A
X
Λn
X
(4)
where
∀,Xj=
x
j
,
1
…
x
j
,
n
j
:X=X1…Xh
Xj
x
j
,
1
…
x
j
,
n
j
X
X1
…
Xh
and
ΛnΛn
is the
nn-by-
nn
diagonal matrix with
λ1λ1
in the first
n1n1
slots,
λ2λ2
in the next
n2n2
slots, etc. Similarly
AAT=YΛmYT
A
A
Y
Λm
Y
(5)
where
∀,Yj=
y
j
,
1
…
y
j
,
n
j
:Y=Y1…Yh
Yj
y
j
,
1
…
y
j
,
n
j
Y
Y1
…
Yh
and
ΛmΛm
is the
mm-by-
mm diagonal matrix with
λ1λ1
in the first
n1n1
slots,
λ2λ2
in the next
n2n2
slots, etc. The
y
j
,
k
y
j
,
k
were defined in
Equation 3
under the assumption that
λj>0
λj
0
. If
λj=0
λj
0
let
YjYj
denote an orthonormal basis for
𝒩AAT
𝒩
A
A
. Finally, call
σj=λj
σj
λj
and let
ΣΣ denote the
mm-by-
nn
matrix diagonal matrix with
σ1σ1
in the first
n1n1
slots,
σ2σ2
in the next
n2n2
slots, etc. Notice that
ΣTΣ=Λn
Σ
Σ
Λn
(6)
ΣΣT=Λm
Σ
Σ
Λm
(7)
Now recognize that
Equation 3 may be
written
A
x
j
,
k
=σj
y
j
,
k
A
x
j
,
k
σj
y
j
,
k
and that this is simply the column by column rendition of
AX=YΣ
A
X
Y
Σ
As
XXT=I
X
X
I
we may multiply through (from the right) by
XT
X
and arrive at the
singular value
decomposition of
AA,
A=YΣXT
A
Y
Σ
X
(8)
Let us confirm this on the
AA matrix in
Equation 1. We have
Λ4=200001000010000
Λ4
2
0
0
0
0
1
0
0
0
0
1
0
0
0
0
X=12-1001020010010020
X
1
2
-1
0
0
1
0
2
0
0
1
0
0
1
0
0
2
0
Λ3=200010001
Λ3
2
0
0
0
1
0
0
0
1
Y=010100001
Y
0
1
0
1
0
0
0
0
1
Hence,
Σ=200001000010
Σ
2
0
0
0
0
1
0
0
0
0
1
0
(9)
and so
A=YΣXT
A
Y
Σ
X
says that
AA should coincide with
010100001200001000010-12012001000001120120
0
1
0
1
0
0
0
0
1
2
0
0
0
0
1
0
0
0
0
1
0
-1
2
0
1
2
0
0
1
0
0
0
0
0
1
1
2
0
1
2
0
This indeed agrees with
AA. It also agrees (up to sign
changes on the columns of
XX) with what one receives upon
typing
[Y, SIG, X] = scd(A) in Matlab.
You now ask what we get for our troubles. I express the first
dividend as a proposition that looks to me like a quantitative
version of the fundamental theorem of linear algebra.
If
YΣXT
Y
Σ
X
is the singular value decomposition of
AA then
-
The rank of AA, call it
rr, is the number of
nonzero elements in
ΣΣ.
-
The first rr columns of
XX
constitute an orthonormal basis for
ℛAT
ℛ
A
. The
n−r
n
r
last columns of XX constitute an
orthonormal basis for
𝒩A
𝒩
A
.
-
The first rr columns of
YY
constitute an orthonormal basis for
ℛA
ℛ
A
. The
m−r
m
r
last columns of YY constitute an
orthonormal basis for
𝒩AT
𝒩
A
.
Let us now 'solve'
Ax=b
A
x
b
with the help of the pseudo-inverse of
AA. You know the 'right' thing to
do, namely reciprocate all of the nonzero singular
values. Because
mm is not
necessarily
nn we must also be
careful with dimensions. To be precise, let
Σ+Σ+
denote the
nn-by-
mm
matrix whose first
n1n1
diagonal elements are
1σ1
1
σ1
, whose next
n2n2
diagonal elements are
1σ2
1
σ2
and so on. In the case that
σh=0
σh
0
, set the final
nhnh
diagonal elements of
Σ+Σ+
to zero. Now, one defines the
pseudo-inverse of
AA to be
A+≡XΣ+YT
A+
X
Σ+
Y
In the case of that
AA is that appearing in
Equation 1 we find
Σ+=200010001000
Σ+
2
0
0
0
1
0
0
0
1
0
0
0
and so
A+=-120120010000011201201200010001000010100001
A+
-1
2
0
1
2
0
0
1
0
0
0
0
0
1
1
2
0
1
2
0
1
2
0
0
0
1
0
0
0
1
0
0
0
0
1
0
1
0
0
0
0
1
therefore,
A+=0-1201000120001
A+
0
-1
2
0
1
0
0
0
1
2
0
0
0
1
in agreement with what appears from
pinv(A). Let us
now investigate the sense in which
A+A+
is the inverse of
AA. Suppose that
b∈ℝm
b
m
and that we wish to solve
Ax=b
A
x
b
. We suspect that
A+b
A+
b
should be a good candidate. Observe by
Equation 4 that
ATAA+b=XΛnXTXΣ+YTb
A
A
A+
b
X
Λn
X
X
Σ+
Y
b
because
XTX=I
X
X
I
ATAA+b=XΛnΣ+YTb
A
A
A+
b
X
Λn
Σ+
Y
b
by
Equation 6 and
Equation 7
ATAA+b=XΣTΣΣ+YTb
A
A
A+
b
X
Σ
Σ
Σ+
Y
b
because
ΣTΣΣ+=ΣT
Σ
Σ
Σ+
Σ
ATAA+b=XΣTYTb
A
A
A+
b
X
Σ
Y
b
by
Equation 8
ATAA+b=ATb
A
A
A+
b
A
b
that is
A+b
A+
b
satisfies the
least-squares problem
ATAx=ATb
A
A
x
A
b
.
