34
CHAPTER 9. EIGENVALUES AND EIGENVECTORS
Answer 9.2
Let
(
λ
,x
)
be an arbitrary eigenpair for
A
, i.e.,
Ax
=
λ
x
.
Then
x
H
Ax
=
λ
x
H
x
. Taking Hermitian transposes of this equation
yields
x
H
A
H
x
=
λ
x
H
x
. Now using the fact that
A
is skewHermitian
we see that
λ
x
H
x
=

λ
x
H
x
. Since
x
is an eigenvector,
x
H
x
±
=0
, from
which we conclude
λ
=

λ
, i.e.,
λ
is pure imaginary.
3. Suppose
A
∈
C
n
×
n
is Hermitian. Let
λ
be an eigenvalue of
A
with
corresponding right eigenvector
x
. Show that
x
is also a left eigenvector
for
λ
. Prove the same result if
A
is skewHermitian.
Answer 9.3
A
=
A
H
,
Ax
=
λ
x
=
⇒
x
H
A
=
x
H
A
H
=
λ
x
H
=
λ
x
H
.
A
=

A
H
,
Ax
=
λ
x
=
⇒
x
H
A
=

x
H
A
H
=

λ
x
H
=
λ
x
H
.
4. Suppose a matrix
A
∈
IR
5
×
5
has eigenvalues
{
2
,
2
,
2
,
2
,
3
}
. Determine
all possible Jordan canonical forms for
A
.
Answer 9.4
The characteristic equation for
A
must be
π
(
λ
) = (
λ

2)
4
(
λ

3)
, thus candidates for the minimal polynomial
α
(
λ
)
are
(
λ

2)(
λ

3)
,
(
λ

2)
2
(
λ

3)
,
(
λ

2)
3
(
λ

3)
, and
(
λ

2)
4
(
λ

3)
. DeFne
the following Jordan blocks for notational ease:
J
4
=
2 1 0 0
0 2 1 0
0 0 2 1
0 0 0 2
,
J
3
=
210
021
002
,
J
2
=
±
21
02
²
,
J
1
=2
, and
J
0
=3
.
There exist Fve possible JC±s for
A
, not counting reorderings of their
diagonal blocks:
diag(
J
4
,J
0
)
,
diag(
J
3
1
0
)
,
diag(
J
2
2
0
)
,
diag(
J
2
1
1
0
)
,
diag(
J
1
1
1
1
0
)
.
5. Determine the eigenvalues, right eigenvectors and right principal vec
tors if necessary, and (real) Jordan canonical forms of the following
matrices:
(a)
±
2

1
10
²
,
(b)
±
12
²
,
(c)
±
45

²
.
(a) Call the matrix
A
. Then
π
(
λ
)=
λ
2

2
λ
+ 1 = (
λ

1)
2
, so
A
has
two eigenvalues at
λ
= 1.
2

rank(
A

I
) = 1, so
A
has one eigenvector and one principal
vector associated with the eigenvalues at 1.