MATH1120, S2 2023
Workshop Solutions Wk 12
Eigenvalues and Eigenvectors
1. Let
A
=
4
0
1
2
3
2

1
0
2
.
Which of the following are eigenvectors for
A
, and
what are their corresponding eigenvalues?
(a) (0
,
1
,
0)
T
(b) (1
,
0
,
1)
T
(c) (

1
,
0
,
1)
T
(d) (0
,
2
,
0)
T
(e) (0
,
0
,
0)
T
(a)
4
0
1
2
3
2

1
0
2
0
1
0
=
0
3
0
= 3
0
1
0
Yes, this is an eigenvector with eigenvalue 3.
(b)
4
0
1
2
3
2

1
0
2
1
0
1
=
5
4

1
No, not an eigenvector.
(c)
4
0
1
2
3
2

1
0
2

1
0
1
=

3
0
3
= 3

1
0
1
Yes, this is an eigenvector with eigenvalue 3.
(d)
4
0
1
2
3
2

1
0
2
0
2
0
=
0
6
0
= 3
0
2
0
Yes, this is an eigenvector with eigenvalue 3.
(e)
4
0
1
2
3
2

1
0
2
0
0
0
=
0
0
0
Even though the zero matrix as input gives the zero matrix as output, it is
not allowed to count as an eigenvector. (Its eigenvalue would not be uniquely
defined.)
Wow, so far all the eigenvalues of this matrix are 3, and we have found only
two linearly independent eigenvectors. Is this all? (Actually, yes, as you could
test with the methods below.)