Matrix Theory, Math6304
Lecture Notes from September 25, 2012
taken by Katie Watkins
Last Time (9/20/12)
•
Cayley Hamilton
•
Block diagonalization with triangular blocks
•
“Nearly” Diagonalizability
Q: We saw Schur does not give a unique upper triangular form. Is there a “nice” choice? How
can we compute it?
2.4
QR Factorization
2.4.1 Theorem
(QR Factorization)
.
Let
A
∈
M
n
,the
reisapa
i
ro
faun
i
ta
ry
Q
and an upper
triangular
R
in
M
n
such that the diagonal entries of
R
are nonnegative and
A
=
QR
,andi
f
A
is invertible, then
R
and
Q
are unique.
Proof.
First, lets examine the invertible case.
Uniqueness:
Let
A
=
Q
1
R
1
=
Q
2
R
2
with
Q
1
,Q
2
∈
M
n
unitary and
R
1
,R
2
upper triangular with nonnegative
diagonal entries
r
(1)
j,j
≥
0
and
r
(2)
j,j
≥
0
.Theen
t
r
ie
sonthed
iagona
lcanno
tbeze
robecau
sethen
we would have an eigenvalue of
0
and then
A
would not be invertible. So
r
(1)
j,j
>
0
and
r
(2)
j,j
>
0
.
We transform the identity to:
Q
1
=
Q
2
R
2
R
−
1
1
Which implies that
M
=
Q
∗
2
Q
1
=
Q
∗
2
(
Q
2
R
2
R
−
1
1
)=
R
2
R
−
1
1
We see that
M
is unitary and upper triangular, which implies that
M
is normal. Since
M
is
upper triangular and normal, we know that it is diagonal. Since the eigenvalues of
R
2
R
−
1
1
are
strictly positive and have magnitude 1, we know that
R
2
R
−
1
1
=
I
=
Q
∗
2
Q
1
.Thu
s
,
R
2
=
R
1
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 Fall '12
 BernhardBodmann
 Math, Determinant, Matrices, Characteristic polynomial, Triangular matrix, upper triangular

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