Day 1 Eigenvalues and Eigenvectors

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.

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?

Answer to Example ASome 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.

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?

Answer

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

The following statements are equivalentA 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

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

Example B answer

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

0 1

1 0

Rotation matrix

What are the eigenvalues an and eigenvectors of a matrix that rotates all vectors 90º?

Recall 2x2 rotation matrices have the form:

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.

How can I solve Ax = λxBring everything on one sideAx – λx = 0(A- λI)x = 0If this can be solved then the matrix(A- λI) must be singularWhich 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)

Find the Eigenvalues3 1

1 3

Find the Eigenvalues

Find det (A- λI) =0 Plug in

(3- λ)2 – 1 λ=2 and find

λ2 - 6 λ + 8 = 0 a basis for

(λ-4)(λ-2)= 0 kernel

λ=4 λ=2

Plug in λ=4 to find the Eigenvectors

find a basis for the null space (kernel)

3 1

1 3

3- λ 1

1 3- λ

-1 1

1 -1

1 1

1 1 Note: this equation is called the characteristic equation

Eigenvalues of triangular matrices

Find the Eigenvalues of

3 1

0 3

Triangular matrices slide 1 of solutions

• Find the Eigenvalues

• A- λ I=

Det(A)= (3 – λ)2 = 0 λ =3

This matrix has a repeated Eigenvalue.

Note: for triangular matrices, the values on the diagonal of the matrix are the Eigenvalues

3- λ 1

0 3 - λ

3 1

0 3 A=

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

0 3 - λ

0 1

0 0

Facts about Eigenvalues

1) An nxn matrix will have n Eigenvalues (values may be repeated)

2) The sum of the Eigenvalues will equal the trace of the matrix

3) 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

Eigenvalues and Eigenvectors on the TI89 Calculator

2nd 5 (math)

4 (matrix)

8 (eigVl)

eigvl([1,2;3,4])

2nd 5 (math)4 (matrix)9 (eigVc)eigVc([1,2;3,4])

1 2

3 4 Find the eigenvalues and eigenvectors on the calculators

Homework (diff 1): worksheet 7.1 5-10 all

textbook p.305 15-21 all