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GramSchmidt and QR
The GramSchmidt process (GSp) takes a sequence
a
1
,a
2
,...,a
n
of linearly independent
vectors and gives a sequence
q
1
,q
2
,...,q
n
of orthonormal vectors which satisfy
Span(
q
1
,q
2
,...,q
k
) = Span(
a
1
,a
2
,...,a
k
)
, k
= 1
,
2
,...,n.
(1)
Everything about the GSp is here in (1). You can even see how it works: Suppose you
have
q
1
,q
2
,...,q
k

1
, and you want
q
k
. Since we want
a
k
∈
Span(
q
1
,q
2
,...,q
k
), we write
a
k
=
r
kk
q
k
+
k

1
X
j
=1
r
jk
q
j
.
(2)
Premultiplying by
q
t
j
(and noting orthogonality) gives
r
jk
=
q
t
j
a
k
, j
= 1
,
2
,...,k

1
,
(3)
and now that the
r
ij
are known we can use (2) to deﬁne
w
≡
r
kk
q
k
=
a
k

k

1
X
j
=1
r
jk
q
j
.
(4)
This gives the direction of
q
k
, and
r
kk
is chosen (usually positive) so that
q
k
has unit
length:
r
kk
=
k
w
k
2
,
(5)
and
q
k
=
w/r
kk
.
(6)
The GSp is simply (3), (4), (5), and (6) for
k
= 1
,
2
,...,n
.
The geometry of the
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This note was uploaded on 12/18/2010 for the course PHYS 5073 taught by Professor Mark during the Fall '10 term at Arkansas.
 Fall '10
 MARK

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