Eigenvectors and eigenvalues
1
1 ,
0
and v
Harold P. Boas
1
1
4
1
1
1
3
1
1
Example. If A
Math 311-102
A non-zero vector v is an eigenvector of a matrix A if Av is a
scalar multiple of v.
2
2
0
boas@tamu.edu
2v.
then Av
Thus v is an eigenvector of A with eigenvalue equal to 2.
How can you find eigenvectors?
The strategy is to find the eigenvalues first.
Math 311-102
June 14, 2005: slide #1
Math 311-102
Finding eigenvalues
Finding eigenvectors
Given a matrix A, the goal is to find a non-zero vector v and a
scalar λ such that Av λv. An equivalent equation is
A λI v 0, where I is the identity matrix.
13 5
has
30 12
3. Find corresponding eigenvectors.
Continuing the example: the matrix A
June 14, 2005: slide #2
eigenvalues 2 and
There can be such a non-zero vector v only if the matrix
A λI fails to be invertible.
June 14, 2005: slide #3
Math 311-102
0 gives
v1
v2
3.
as an eigenvector with eigenvalue
5
15
10
30
1
2
0 or
3I v
1
.
3
1
.
3
2
0 or
3.
λ
150
0 or λ2 λ 6
0. The eigenvalues are 2 and
2
6
1
3
Similarly solving A
Math 311-102
λ
λ 12
3 λ 2
13
λ
v1
v2
0, so
12
5
12
v1
v2
0 or
5
Solution. Solve det
Check:
13 λ
30
1
0
13
30
reducing gives
5
.
12
3
0
13
30
Example. Find the eigenvalues of
0 determines the eigenvalues; it is
The equation det A λI
called the characteristic equation of the matrix A.
Solution. The eigenvector v corresponding to eigenvalue 2
15 5
v1
satisfies A 2I v 0 or
0. Row
30 10
v2
June 14, 2005: slide #4
Change of basis
5
12
13
30
The matrix A
represents a linear transformation
0
3
2
0
Answer. The diagonal matrix D
entries are the eigenvalues.
1 , 0
.
with respect to the standard basis
0
1
What matrix represents the same transformation with respect
1 , 1
to the basis
of eigenvectors?
3
2
whose diagonal
and U
1 AU
D. (Check!)
1
UDU
Then A
1 1 (whose columns are the eigenvectors)
The matrix U
32
translates from eigenvector coordinates to standard
2 1
coordinates. The matrix U 1
3 1 translates from
standard coordinates to eigenvector coordinates.
We can diagonalize a matrix by using a basis of eigenvectors.
Math 311-102
June 14, 2005: slide #5