1. Day 1 Eigenvalues and Eigenvectors

2. Suppose we have some vector A, in the equation Ax=b and we want to find the which vectors x are pointing the same direction after the transformation. These vectors are called Eigenvectors.
The vector b must be a scalar multiple of x. The scalar that multiplies x is called the Eigenvalue
The main equation for this section is
Ax = λx
Any vector x that satisfies this equation is an Eigenvector, the corresponding λ is the Eigenvalue
Note: for this section we are only considering square matrices.

3. Example A
Let’s examine some vectors that we are already familiar with and determine the Eigenvectors and Eigenvalues.
Consider a Projection matrix P in R3, that projects vectors on to a plane. What are the Eigenvectors and Eigenvalues?

4. Answer to Example A
Some Eigenvectors are the vectors that are already in the plane that is being projected on. In that case the vector does not change so the Eigenvalue for these vectors is 1
Other Eigenvectors are those orthogonal to the plane that is being projected on. Those vectors become the zero vector (which is considered parallel to all vectors). The Eigen value for these vectors is zero.

5. Look at the case λ = 0 If A is a singular matrix, then we can solve
Ax = λx
What did we previously call these values?

6. Answer
If λ= 0 then we are solving Ax=0 which is the null space (Kernel)

7. The following statements are equivalent
A is invertible
The linear system Ax=b has a unique solution x for all b
rref(A) = In
Rank(A) = n
Im (A) = Rn
ker(A) = 0
The column vectors of A form a basis of Rn
The column vectors of A span Rn
The column vectors of A are linearly independent
detA ≠0
0 fails to be an eigenvalue of A

8. Example B Permutation Matrix What does this vector do to the x’s?
What is a vector with λ =1?
What is a vector with λ = -1?
0 1
1 0

9. Example B answer Permutation Matrix
0 1
1 0
Permutation Matrix
What does this vector do to the x’s? (changes the order of the components of a vector)
What is a vector with λ =1? [1;1] any with repeated values
What is a vector with λ = -1? [-1;1] any with opposite values

10. Rotation matrix
What are the eigenvalues an and eigenvectors of a matrix that rotates all vectors 90º?
Recall 2x2 rotation matrices have the form:

11. Rotation matrix
There will not be any real Eigenvalues or vectors. (the eigenvalues will be imaginary)
Rotation matrix rotate all vectors so no real vectors will come out of the system in the direction that they go in.
Note: zero can be an eigenvalue but it can not be an eigenvector.

12. How can I solve Ax = λx Bring everything on one side Ax – λx = 0
(A- λI)x = 0
If this can be solved then the matrix
(A- λI) must be singular
Which means that det (A- λI) =0
This equation is called the characteristic equation.
There should be n values to this equation (although some could be repeated)
Once we find λ find the nullspace of(A- λI)x = 0
to find the x’s (Eigenvectors)

13. 3 1
1 3
Find the Eigenvalues

14. Find the Eigenvalues Find det (A- λI) =0 Plug in
3 1
1 3
Find the Eigenvalues
1 1
1 1
Note: this equation is
called the characteristic
equation
Find det (A- λI) = Plug in
(3- λ)2 – λ=2 and find
λ2 - 6 λ + 8 = a basis for
(λ-4)(λ-2)= kernel
λ=4 λ=2
Plug in λ=4 to find the Eigenvectors
find a basis for the null space (kernel)
3- λ 1 λ
-1 1

15. Eigenvalues of triangular matrices
Find the Eigenvalues of
3 1
0 3

16. Triangular matrices slide 1 of solutions
3 1
0 3
Find the Eigenvalues
A- λ I=
Det(A)= (3 – λ)2 = λ =3
This matrix has a repeated Eigenvalue.
Note: for triangular matrices, the values on the diagonal of the matrix are the Eigenvalues A=
3- λ 1 λ

17. Triangular matrices Find the Eigenvectors A- λ I= Replace λ by 3
Find the null space
This matrix has only 1 Eigenvector!
A repeated λ gives the possibility of a lack of Eigenvectors
3- λ 1 λ

18. Facts about Eigenvalues
An nxn matrix will have n Eigenvalues (values may be repeated)
The sum of the Eigenvalues will equal the trace of the matrix
The product of the eigenvalues will be the determinant of the matrix
Note: a Trace is the sum of the numbers on the diagonal of the matrix

19. Eigenvalues and Eigenvectors on the TI89 Calculator
1 2
3 4
Find the eigenvalues and eigenvectors on the calculators
2nd 5 (math)
4 (matrix)
8 (eigVl)
eigvl([1,2;3,4])
2nd 5 (math)
4 (matrix)
9 (eigVc)
eigVc([1,2;3,4])

20. Homework (diff 1): worksheet 7.1 5-10 all textbook p.305 15-21 all