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Homework 5 Numerical Linear Algebra 1 Fall 2010
Due date: beginning of class Wednesday, 11/5/10
Problem 5.1
Let
A
∈
R
n
×
k
have rank
k
. The pseudoinverse for rectangular full columnrank matrices
behaves much as the inverse for nonsingular matrices. To see this show the following identities
are true (Stewart 73):
5.1.a
.
AA
†
A
=
A
5.1.b
.
A
†
AA
†
=
A
†
5.1.c
.
A
†
A
= (
A
†
A
)
T
5.1.d
.
AA
†
= (
AA
†
)
T
5.1.e
. If
A
∈
R
n
×
k
has orthonormal columns then
A
†
=
A
T
. Why is this important for
consistency with simpler forms of least squares problems that we have discussed?
Problem 5.2
Any subspace
S
of
R
n
of dimension
k
≤
n
must have at least one orthogonal matrix
Q
∈
R
n
×
k
with orthonormal columns such that
R
(
Q
) =
S
, The matrix
P
=
QQ
T
is a projector onto
S
, i.e.,
Px
is the unique component of
x
contained in
S
.
5.2.a
.
P
is clearly symmetric, show that it is idempotent, i.e.,
P
2
=
P
.
5.2.b
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 Fall '06
 gallivan

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