ENEE 241 02*
READING ASSIGNMENT 10
Tue 03/10 Lecture
Topics:
solution of the projection (least squares approximation) problem; complexvalued vectors
Textbook References:
sections 2.10.4–2.10.5, 2.13
Key Points:
•
The projection of a
m
dimensional vector
s
onto the range
R
(
V
) of a matrix
V
is the unique
vector ˆ
s
in
R
(
V
) with the property that
s
−
ˆ
s
is orthogonal to every column of
V
. It is also
the point in
R
(
V
) closest to
s
.
•
ˆ
s
is obtained by solving the
n
×
n
system of equations
(
V
T
V
)
c
=
V
T
s
for
c
, followed by forming the linear combination ˆ
s
=
Vc
. The coefficient vector
c
is unique
if the columns of the matrix
V
are linearly independent.
•
For complex vectors, the inner product and norm are defined by
(
v
,
w
)
=
m
summationdisplay
i
=1
v
*
i
w
i
= (
v
*
)
T
w
bardbl
v
bardbl
=
(
v
,
v
)
1
/
2
=
parenleftBigg
m
summationdisplay
i
=1

v
i

2
parenrightBigg
1
/
2
(
v
*
)
T
is also denoted as
v
H
.
•
The projection of a complexvalued vector
s
onto the range of a complexvalued matrix
V
is
given by
Vc
, where
(
V
H
V
)
c
=
V
H
s
Theory and Examples:
1
. We saw that the projection ˆ
s
of a
m
dimensional vector
s
onto the range
R
(
V
) of a matrix
V
is characterized by two equivalent properties:
• bardbl
s
−
ˆ
s
bardbl ≤ bardbl
s
−
y
bardbl
for any
y
∈ R
(
V
), with equality only when
y
= ˆ
s
;
•
s
−
ˆ
s
is orthogonal to every column of
V
.
Since ˆ
s
=
Vc
for some
c
∈
R
n
×
1
, the latter property is also expressed as
(
V
T
V
)
c
=
V
T
s
2
.
V
T
V
is a symmetric
n
×
n
matrix.
Its (
i, j
)
th
entry equals the inner product
(
v
(
i
)
,
v
(
j
)
)
.
Note also that this matrix does not involve
s
in any way.
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 Spring '08
 staff
 Linear Algebra, Vector Space

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