This preview shows pages 1–4. Sign up to view the full content.
Economics 204
Lecture 10–Friday, August 7, 2009
Revised 8/8/09, Revisions indicated by ** and
Sticky Notes
Diagonalization of Symmetric Real Matrices (from
Handout):
De±nition 1
Let
δ
ij
=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
1i
f
i
=
j
0i
f
i
6
=
j
Abas
is
V
=
{
v
1
,...,v
n
}
of
R
n
is
orthonormal
if
v
i
·
v
j
=
δ
ij
.In
other words, each basis element has unit length, and distinct basis
elements are perpendicular.
Observation
: Suppose that
x
=
∑
n
j
=1
α
j
v
j
where
{
v
1
n
}
is
an orthonormal basis of
V
. Then for any
x
∈
V
,
x
·
v
k
=
⎛
⎜
⎝
n
±
j
=1
α
j
v
j
⎞
⎟
⎠
·
v
k
=
n
±
j
=1
α
j
(
v
j
·
v
k
)
=
n
±
j
=1
α
j
δ
jk
=
α
k
so
x
=
n
±
j
=1
(
x
·
v
j
)
v
j
Example:
The standard basis of
R
n
is orthonormal.
De±nition 2
Area
l
n
×
n
matrix
A
is
unitary
if
A
>
=
A
−
1
,
where
A
>
denotes the transpose of
A
:th
e(
i, j
)
th
entry of
A
>
is
the (
j, i
)
th
entry of
A
.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document Theorem 3
Area
l
n
×
n
matrix
A
is unitary if and only if
the columns of
A
are orthonormal.
Proof:
Let **
v
j
denote the
j
th
column of
A
.
A
>
=
A
−
1
⇔
A
>
A
=
I
⇔
v
i
·
v
j
=
δ
ij
⇔{
v
1
,...,v
n
}
is orthonormal
If
A
is unitary, let
V
be the set of columns of
A
and
W
be the
standard basis of
R
n
.
Since
A
is unitary, it is invertible, so
V
is a basis of
R
n
.
A
>
=
A
−
1
=
Mtx
V,W
(
id
)
Since
V
is orthonormal, the transformation between bases
W
and
V
preserves all geometry, including lengths and angles.
Theorem 4
Let
T
∈
L
(
R
n
,
R
n
)
,
W
the standard basis of
R
n
. Suppose that
W
(
T
)
is symmetric. Then the eigen
vectors of
T
are all real, and there is an orthonormal basis
V
=
{
v
1
n
}
of
R
n
consisting of eigenvectors of
T
,s
o
that
W
(
T
)
is diagonalizable:
W
(
T
)
=
W,V
(
id
)
·
V
(
T
)
·
(
id
)
where
V
T
is diagonal and the change of basis matrices
(
id
)
and
(
id
)
are unitary.
The proof of the theorem requires a lengthy digression into the
linear algebra of
complex
vector spaces. Here is a very brief out
line.
2
1. Let
M
=
Mtx
W
(
T
).
2. The inner product in
C
n
is defned as Follows:
x
·
y
=
n
±
j
=1
x
j
·
y
j
where ¯
c
denotes the complex conjugate oF any
c
∈
C
;no
t
e
that this implies that
x
·
y
=
y
·
x
. The usual inner product
in
R
n
is the restriction oF this inner product on
C
n
to
R
n
.
3. Given any complex matrix
A
, defne
A
∗
to be the matrix whose
(
i, j
)
th
entry is
a
ji
; in other words,
A
∗
is Formed by taking the
complex conjugate oF each element oF the transpose oF
A
.I
t
is easy to veriFy that given
x, y
∈
C
n
and a complex
n
×
n
matrix
A
,
Ax
·
y
=
x
·
A
∗
y
.S
in
c
e
M
is real and symmetric,
M
∗
=
M
.
4. IF **
M
is real and symmetric, and
λ
∈
C
is an eigenvalue oF
M
, with eigenvector
x
∈
C
n
,then
λ

x

2
=
λ
(
x
·
x
)
=(
λx
)
·
x
Mx
)
·
x
=
x
·
(
M
∗
x
)
=
x
·
(
)
=
x
·
(
λx
)
=
(
λx
)
·
x
=
λ
(
x
·
x
)
=
λ

x

2
=
¯
λ

x

2
which proves that
λ
=
¯
λ
, hence
λ
∈
R
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 10/04/2010 for the course ECON 204 taught by Professor Anderson during the Summer '08 term at University of California, Berkeley.
 Summer '08
 ANDERSON
 Economics

Click to edit the document details