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5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES
Figure 5
The equation can be rearranged as
(5.9)
p
=
a
μ
a
T
y
a
T
a
¶
=
∙
1
a
T
a
¸
£
aa
T
¤
y
=
P
r
.
y
where
P
r
=
1
a
T
a
aa
T
is a
n
×
n
matrix is called as
projection matrix,
which
projects vector
y
into its column space.
5.2. Distance of a point from Subspace.
The situation is exactly
same when we are given a point
y
∈
R
3
and plane
S
in
R
3
passing through
origin
,
we want to
f
nd distance of
y
from
S,
i.e. a point
p
∈
S
such that
k
p
−
y
k
2
is minimum (see Figure 5). Again, from school geometry, we know that such
point can be obtained by drawing a perpendicular from
y
to
S
;
p
is the point
where this perpendicular meets
S
(see Figure 5). We would like to formally
derive this result using optimization.
More generally, given a point
y
∈
R
m
and subspace
S
of
R
m
,thep
rob
lem
is to
f
nd a point
p
in subspace
S
such that it is closest to vector
y
.L
e
t
S
=
span
©
a
(1)
,
a
(2)
, ....
,
a
(
m
)
ª
and as
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This note was uploaded on 11/26/2011 for the course EGN 3840 taught by Professor Mr.shaw during the Fall '11 term at FSU.
 Fall '11
 Mr.Shaw
 Chemical Engineering, Optimization

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