Math 235
Assignment 9 Solutions
1.
Suppose that a real 2
×
2 matrix
A
has 2 +
i
as an eigenvalue with a corresponding
eigenvector
±
1 +
i
i
²
. Determine
A
.
Solution: Since
A
is real, we know that
A
has real canonical form
B
=
±
2
1

1 2
²
and is
brought into this form by
P
=
±
1 1
0 1
²
. We then have that
A
=
PBP

1
=
±
1 1
0 1
²±
2
1

1 2
²±
1

1
0
1
²
=
±
1
2

1 3
²
.
2.
Determine a real canonical form of
A
=
0

2
1
2
2

1
0

2
2
and give a change of basis matrix
P
that brings the matrix into this form.
Solution: We have
C
(
λ
) =

(
λ

2)(
λ
2

2
λ
+ 2). The roots are 2 and 1
±
i
, so the
eigenvalues of
A
are
μ
= 2,
λ
= 1 +
i
and
λ
. Hence, a real canonical form is
2
0
0
0
1
1
0

1 1
.
For
μ
= 2 we get
A

μI
=

2

2
1
2
0

1
0

2
0
∼
1 0

1
/
2
0 1
0
0 0
0
.
So, an eigenvector corresponding to
μ
is
~v
=
1
0
2
.
For
λ
= 1 +
i
we get
A

λI
=

1

i

2
1
2
1

i

1
0

2
1

i
∼
1 0

1
2

1
2
i
0 1

1
2
+
1
2
i
0 0
0
.
So, an eigenvector corresponding to
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 Spring '10
 WILKIE
 Math, Linear Algebra, λ, eigenvector, real canonical form

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