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
, where
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
This preview has intentionally blurred sections. Sign up to view the full version.
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
∗
XU
)
=
U
∗
X
∗
UU
∗
XU
=
U
∗
X
∗
XU
=
U
∗
XX
∗
U
=
U
∗
XUU
∗
X
∗
U
=
Y Y
∗
,
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
∞
a diagonal matrix and
A
∞
=
D
∞
Q
∞
=
D
∞
Q
∞
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '12
 BernhardBodmann
 Linear Algebra, Addition, Matrices, Triangular matrix, Orthogonal matrix

Click to edit the document details