In each of the following cases, let
T
be the linear operator on
R
2
which is represented by the matrix
A
in the standard ordered basis for
R
2
, and
let
U
be the linear operator on
C
2
represented by
A
in the standard ordered basis.
Find the characteristic polynomial for
T
and that for
U
, find the characteristic
values of each operator, and for each such characteristic value
c
find a basis for
the corresponding space of characteristic vectors.
A
=
1
0
0
0
¶
A
=
2
3

1
1
¶
A
=
1
1
1
1
¶
In all cases, denote by
B
c
the basis for the subspace corresponding
to the characteristic value
c
First matrix The characteristic polynomial is
x
(
x

1)
. The roots are
0
and
1
.
B
0
=
{
(0
,
1)
}
and
B
1
=
{
(1
,
0)
}
. In this case the real and complex cases are the same.
Second matrix The characteristic polynomial is
5

3
x
+
x
2
which has no real roots. The
complex eigenvalues are
3
2
+
i
1
2
√
11
,
3
2

i
1
2
√
11
,
B
3
2
+
i
1
2
√
11
=
{
(1
,

1
6
+
i
1
6
√
11)
}
and
B
3
2

i
1
2
√
11
=
{
(1
,

1
6

i
1
6
√
11)
}
Third matrix The characteristic polynomial is
x
2

2
x
. The roots are
0
and
2
.
B
0
=
{
(

1
,
1)
}
and
B
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.
 Spring '10
 aa
 Linear Algebra, Complex number, characteristic value

Click to edit the document details