 # Day 1 Eigenvalues and Eigenvectors

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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:

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 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

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

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

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 kernel (λ-4) (λ-2) = 0 λ = 4 and λ = 2 Plug in λ = 4 to find the Eigenvectors find a basis for the null space (kernel) 3- λ 1 λ -1 1

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

Triangular matrices slide 1 of solutions
3 1 0 3 Find the Eigenvalues A- λI = 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

Triangular matrices Find the Eigenvectors A- λI = 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

An n x n 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.