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
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
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
=
w
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
k
th
step of GSp is simple:
For each
j
= 1
,
2
, . . . , k

1, (3) and (4) subtracts from
a
k
its projection onto
q
j
. The
resulting vector is then orthogonal to
q
j
. The final vector,
w
, is then orthogonal to
q
1
, q
2
, . . . , q
k

1
. Interpret
r
jk
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 Fall '10
 MARK
 Linear Algebra, QR factorization, Qk

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