Matrix Theory, Math6304
Lecture Notes from October 9, 2012
taken by Charles Mills
Last Time (10/6/12)
Suppose
A
∈
M
n
satisfies
A
=
A
∗
, then given
A
, we can define a map
q
A
:
C
n
→
R
by
q
A
(
x
) =
< Ax, x > .
It is clear that if we know the matrix
A
, we know
q
A
and vice versa. This
section is about obtaining information about the eigenvalues of
A
from
q
A
.
4
Variational characterization of eigenvalues
4.1
RayleighRitz
We introduce some notation.
In the following section, we always order the eigenvalues of a
Hermitian
A
such that
λ
min
=
λ
1
≤
λ
2
≤
· · ·
≤
λ
n
=
λ
max
,
multiplicity counted.
4.1.1 Theorem.
(RayleighRitz) Let
A
∈
C
n
satsify
A
=
A
∗
, then:
1.
∀
x
∈
C
n
,
λ
min

x

2
≤
< Ax, x >
≤
λ
max

x

2
.
2.
•
λ
max
=
max

x

=1
< Ax, x >
=
max
x
=0
<Ax,x>

x

2
•
λ
min
=
min

x

=1
< Ax, x >
=
min
x
=0
<Ax,x>

x

2
Proof.
Since
A
=
A
∗
,
A
is diagonalizable by a unitary matrix
U
. Let
U
be such that
•
UAU
∗
=
D
•
D
is diagonal with
d
ii
=
λ
i
•
U
∗
U
=
I
•
U
= [
U
1
, U
2
, . . . , U
n
]
where
AU
i
=
λ
i
U
i
1. Given such an
A, U,
and
D
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 Fall '12
 BernhardBodmann
 Math, Matrices, Trigraph, ax, CN, Orthogonal matrix

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