Digital Signal Processing Reference
In-Depth Information
V
[
Σ0
]
Σ
0
V
†
=
VΣ
2
V
†
.
=
From this we obtain
Tr (
A
†
A
)=Tr
VΣ
2
V
†
)
=Tr
V
†
VΣ
2
)
2
)=
k
σ
k
,
=Tr
Σ
which proves the claim. In the second equality above we have used the trace
identity, which says that, Tr (
PQ
)=Tr(
QP
)whenver
PQ
and
QP
are
both defined.
C.4 Frobenius norm of the left inverse
For the
P
×
M
matrix
A
with rank
M
the SVD is
Σ
0
V
†
,
A
=
U
(C
.
12)
and a valid left inverse is
A
#
=
V
[
Σ
−
1
0
]
U
†
.
(C
.
13)
The Frobenius norm of the left inverse
A
#
can be calculated by observing that
A
#
2
=Tr
A
#
[
A
#
]
†
.
(C
.
14)
A
#
2
=
M−
1
k
=0
1
/σ
k
.
Summarizing, the Frobenius norms of
A
and its left inverse
A
#
are given by
Proceeding as in the proof of Eq. (C.11) this simplifies to
M−
1
M−
1
1
σ
k
A
2
=
σ
k
,
A
#
2
=
.
(C
.
15)
k
=0
k
=0
Rank-deficient case.
For the case where
A
does not have full rank we can use
the representation (C.3) for
A
and the representation (C.8) for the pseudoinverse
A
#
,where
ρ
is the rank of
A
(and of
A
#
). In this case the expressions (C.15)
continue to hold with the modification that both the summations have the range
0
≤ k ≤ ρ −
1
.
Minimum-norm left inverse
If
A
M
with rank
M,
and if
P>M,
then there are infinitely many left
inverses, and
A
#
is just one of them. To see this, observe that there are
P
is
P
×
−
M
linearly independent nonzero vectors
v
such that
v
†
A
=
0
.
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