Econ 204
Supplement to Section 3.6
Diagonalization and Quadratic Forms
De La Fuente notes that, if an
n
×
n
matrix has
n
distinct eigenvalues,
it can be diagonalized. In this supplement, we will provide an additional,
very important diagonalization result.
Symmetric matrices can always be
diagonalized; moreover, the change of basis matrices that carry out the di
agonalization have a special form.
This has an important application to
quadratic forms, which in turn have application to the geometry of level sets
of preferences, and to the analysis of variancecovariance matrices.
1
Diagonalization and Change of Basis
Before proceeding with the diagonalization result for symmetric matrices, it is
useful to discuss the relationship between diagonalization and change of basis.
De La Fuente (page 151) de±nes a square matrix
M
to be
diagonalizable
if
there exists an invertible matrix
P
such that
P
−
1
MP
is diagonal; he also
(page 146) de±nes two square matrices
A
and
B
to be
similar
if there is
an invertible matrix
P
such that
P
−
1
AP
=
B
,soasqua
r
ema
t
r
ix
M
is
diagonalizable if and only if it is similar to a diagonal matrix. Theorem 2
tells us that a matrix is diagonalizable if and only if there is another basis
so that the representation of the same transformation in the new basis is
diagonal.
Proposition 1
Fix an
n
dimensional vector space
X
and a basis
U
=
{
u
1
,...,u
n
}
.
An
n
×
n
matrix
P
is invertible if and only if there is a basis
W
such that
P
=(
Mtx
)
U,W
(
id
)
; in this case,
((
)
U,W
(
id
))
−
1
)
W,U
(
id
)
.
Proof:
Suppose that
P
is invertible. Let
W
=
{
w
1
,...,w
n
}
,whe
re
w
j
=
∑
n
i
=1
p
ij
u
i
.S
i
n
c
e
P
is invertible, rank
P
=
n
,s
o
W
is a basis.
P
=
(
)
U,W
(
id
).
Conversely, suppose there is a basis
W
such that
P
)
U,W
(
id
).
Then
w
j
=
∑
n
i
=1
p
ij
u
i
.
By the Commutative Diagram Theorem (see the
1
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View Full DocumentSupplement to Section 3.3),
(
Mtx
)
U,W
(
id
)
·
(
)
W,U
(
id
)=(
)
U,U
(
id
◦
id
)
=(
)
U
(
id
)
=
I
so
P
is invertible and (
)
W,U
(
id
)=((
)
U,W
(
id
))
−
1
.
Theorem 2
Suppose that
X
is fnitedimensional
•
IF
T
∈
L
(
X, X
)
,and
U, W
are any two bases oF
X
,then
(
)
W
(
T
)
and
(
)
U
(
T
)
are similar.
•
Conversely, given similar matrices
A, B
with
A
=
P
−
1
BP
and any
basis
U
, there is a basis
W
and
T
∈
L
(
X, X
)
such that
B
)
U
(
T
)
,
A
)
W
(
T
)
,
P
)
U,W
(
id
)
P
−
1
)
W,U
(
id
)
.
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 Fall '08
 ANDERSON
 Linear Algebra, MTx, Commutative Diagram Theorem

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