Matrix Theory, Math6304
Lecture Notes from September 27, 2012
taken by Tasadduk Chowdhury
Last Time (09/25/12):
QR factorization:
any matrix
A
∈
M
n
has a QR factorization:
A
=
QR
,whe
re
Q
is unitary
and
R
is upper triangular. In addition, we proved that if
A
is nonsingular,
Q
and
R
are
unique.
Cholesky factorization:
any
B
∈
M
n
satisfying
B
=
A
∗
A
,
A
∈
M
n
has the factorization
B
=
LL
∗
where
L
∈
M
n
is lower triangular and nonnegative on the diagonal.
QR algorithm:
details and convergence criteria.
Warmup
If
A
=
A
0
∈
M
n
is normal and QR algorithm converges (entrywise), what can we say about the
limiting matrix
A
∞
?
Convergence implies that there is a unitary
Q
∞
and a uppertriangular
R
∞
such that
A
∞
=
Q
∞
R
∞
=
R
∞
Q
∞
.
(1)
From above, we see that
Q
∞
and
R
∞
commute. Also,
A
∞
Q
∞
=
Q
∞
R
∞
Q
∞
=
Q
∞
A
∞
.
(2)
So
Q
∞
and
A
∞
also commute. Moreover, from (1) we get
Q
∗
∞
A
∞
=
R
∞
=
R
∞
Q
∞
Q
∗
∞
=
A
∞
Q
∗
∞
.
(3)
By taking adjoints, we get
A
∗
∞
Q
∗
∞
=
Q
∗
∞
A
∗
∞
from (2) and
Q
∞
A
∗
∞
=
A
∗
∞
Q
∞
from (3). Thus,
R
∞
R
∗
∞
=
Q
∗
∞
A
∞
A
∗
∞
Q
∞
=
A
∞
Q
∗
∞
Q
∞
A
∗
∞
(by commutativity)
=
A
∞
A
∗
∞
=
A
∗
∞
A
∞
(by normality of
A
∞
)
=
A
∗
∞
Q
∞
Q
∗
∞
A
∞
=
Q
∞
A
∗
∞
A
∞
Q
∗
∞
(by commutativity)
=
R
∗
∞
R
∞
.
1
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View Full Document(We claimed above that
A
∞
is normal. From last lecture’s proposition of QR algorithm we know
that
A
n
’s are all unitarily equivalent. It can be checked that any matrix unitarily equivalent to a
normal matrix is also normal: say
X
and
Y
are unitarily equivalent, and
X
is normal. Then,
Y
∗
Y
=(
U
∗
XU
)
∗
(
U
∗
)
=
U
∗
X
∗
UU
∗
=
U
∗
X
∗
=
U
∗
XX
∗
U
=
U
∗
XUU
∗
X
∗
U
=
YY
∗
,
and thus
Y
is normal. Hence, since
A
n
’s are unitarily equivalent to
A
0
=
A
,and
A
is normal,
and thus the limiting matrix
A
∞
is also normal.)
By normality of
R
∞
and the fact that it is triangular,
R
∞
=
D
∞
with
D
∞
ad
iagona
lmatr
ixand
A
∞
=
D
∞
Q
∞
=
D
∞
Q
∞
.
Since
{
Q
∞
,D
∞
}
is a commuting family in
M
n
,thereex
istsaun
ita
ry
U
∈
M
n
that diagonalizes
both:
U
∗
D
∞
U
=
D
∞
,
and
U
∗
Q
∞
U
=
°
Q
∞
,
where
°
Q
∞
is a diagonal matrix. We will denote the
i
’th diagonal entry of of
°
Q
∞
by
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 Fall '12
 BernhardBodmann
 Linear Algebra, Addition, Matrices, Triangular matrix, Orthogonal matrix

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