Day 1 Eigenvalues and Eigenvectors
Suppose we have some vector A, in the equation Ax=b
and we want to find which vectors x are pointing in
the same direction (parallel) 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 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
Eigenvalue for these vectors is zero.
Singular Matrix
A square matrix that does not have a matrix
inverse. A matrix is singular iff its
determinant is 0. For example, there are 10
singular 2×2 (0,1)-matrices:
http://mathworld.wolfram.com/SingularMatrix.html
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 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
det A ≠ 0
0 fails to be an eigenvalue of A
Example B
Permutation Matrix
0
1
1 0
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
Example B answer
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
Rotation matrix
What are the eigenvalues 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.
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)
3 1
1
3
Find the Eigenvalues
3 1
1
3
Find the Eigenvalues
1 1
1
Note: this equation is
called the characteristic
equation
1
Find det (A- λI) = 0
Plug in
(3- λ)2 – 1
λ = 2 and find
3- λ 1
λ2 - 6 λ + 8 = 0
a basis for kernel
1 3- λ
(λ-4) (λ-2) = 0
λ = 4 and λ = 2
-1 1
Plug in λ = 4 to find the Eigenvectors
1 -1
find a basis for the null space (kernel)
Eigenvalues of triangular matrices
Find the Eigenvalues of
3
1
0
3
Triangular matrices slide 1 of solutions
• Find the Eigenvalues
• A- λI =
3- λ
0
3 1
A=
0
3
1
3-λ
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.
Triangular matrices
Find the Eigenvectors A- λI =
3- λ
0
Replace λ by 3
Find the null space
0
1
0
0
1
3-λ
This matrix has only 1 Eigenvector!
A repeated λ gives the possibility of a lack of
Eigenvectors
Facts about Eigenvalues
1) An n x n 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.
More info on eigenvalues &
eigenvectors
http://mathworld.wolfram.com/Eigenvalue.html
http://mathworld.wolfram.com/Eigenvector.html
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])
Homework: worksheet 7.1 5-10 all
textbook p.305 15-21 all