Matrix Theory, Math6304
Lecture Notes from September 18, 2012
taken by John Haas
Last Time (9/13/12)
Unitary Matrices:
defnition and some characterizations/results
Householder Transforms
:ba
s
i
ccompu
ta
t
ion
Unitary Equivalence:
defnition and some conditions
ClariFcation
IF
A
∈
M
n
(
C
)
,then
:
°
Ax, x
±
=0
∀
x
∈
C
n
i±
A
.
However, in the case where
F
=
R
,werequ
ireanextracond
it
:
IF
A
∈
M
n
(
R
)
and
A
=
A
∗
:
°
Ax, x
±
∀
x
∈
R
n
i±
A
.
Warmup
Suppose
A
∈
M
n
has orthogonal columns, all oF which are nonzero. Can we compute
A
−
1
easily?
²rom our characterizations oF unitary matrices From last time, we know we can write:
[
A
]
ij
=
d
i
u
ij
,where
U
=(
u
ij
)
is unitary and
d
i
²
.
Hence, by using our knowledge oF right multiplication with diagonal matrices, we get:
A=UD, where
d
i
is the ith entry along the the diagonal oF the diagonal matrix D.
So it Follows that:
A
−
1
=
D
−
1
U
−
1
=
D
−
1
U
∗
=
1
d
1
0
···
0
0
1
d
2
0
00
.
.
.
0
1
d
n
U
∗
1
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View Full DocumentThus, using our knowledge of left multiplication with diagonal matrices, we get:
[
A
−
1
]
ij
=
d
−
1
i
[
U
∗
]
ij
=
d
−
1
i
¯
u
ji
2
Schur Triangularization and Consequesnces (continued)
2.1
Schur’s triangularization theorem
2.1.1 Theorem.
(Schur’s Triangularization Lemma)
Let
A
∈
M
n
,w
i
the
igenva
lue
s
λ
1
,λ
2
,...,λ
n
(where multiplicity is counted). Then there exists
U
∈
M
n
,where
U
∗
U
=
I
,suchthat
:
A
=
UTU
∗
and
T
=
λ
1
∗·
·
· ∗
0
λ
2
···
∗
00
.
.
.
∗
λ
n
is upper triangular.
In other words, every square matrix is unitarily equivalent to an upper triangular matrix.
Proof.
The proof follows by induction on the dimension, n:
(
n
=1):
Done.
(
n
−
1
⇒
n
):
Suppose Schur’s theorem holds for all
(
n
−
1)
×
(
n
−
1)
matrices, and let
A
∈
M
n
,
with eigenvalues
λ
1
2
n
.C
h
o
o
s
ea
ne
i
g
e
n
v
a
l
u
e
x
corresponding to
λ
1
and, WLOG, sup
pose

x

=1
.Nowe
x
t
end
{
x
}
to a basis for
C
n
,andthenapp
lytheG
ram
Schm
idttogetan
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 Fall '12
 BernhardBodmann
 Matrices, Diagonal matrix, Triangular matrix, Normal matrix, upper triangular

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