SLC Math 54 Adjunct
Faculty:
Prof. Srivastava
Instructor:
Brittnee Mazorra, [email protected]
Location:
TuTh 23:30pm, Zoom
Office Hours:
TBD in
SLC Virtual DropIn
Worksheet 16: Complex Eigenstuff (5.5))
Complex Eigenstuff
1. Consider the matrix
A
=
0

1
1
0
.
(a) Show
A
2
=

I
.
(b) Suppose
λ
is an eigenvalue for
A
corresponding to
x
. Show that
λ
=
±
i
using the fact that
A
2
=

I
. (we
could of course use determinants to show this as well)
(c) Find the eigenvectors of
A
.
2. For each of the following matrices, find all eigenvalues and bases for the corresponding eigenspaces.
(a)
3

4
4
3
(b)
1

2
3
1
(c)
1

1

1
1

1
0
1
0

1
3. Show that if
a, b
∈
R
, then the eigenvalues of
A
=
a

b
b
a
are
a
±
bi
, with corresponding eigenvectors
i
1
,

i
1
.
4. Find an invertible matrix
P
and a matrix
C
of the form
a

b
b
a
such that
A
=
PCP

1
.
(a)
A
=
3

4
4
3
(b)
A
=
1

2
3
1
(c)
A
=

1

2
1
1
(d)
A
=
3

2
1
3
5. Many properties of complex numbers carry over to complex matrix algebra.
For example, if
A
∈
R
n
×
n
then
A
x
=
A
x
. Show that if