2 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 = λxAny 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 ALet’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 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 Eigenvalue for these vectors is zero.
5 Singular MatrixA 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:
6 Look at the case λ = 0 If A is a singular matrix, then we can solve Ax = λxWhat did we previously call these values?
7 AnswerIf λ = 0 then we are solving Ax = 0 which is the null space (Kernel)
8 The following statements are equivalent A is invertibleThe linear system Ax = b has a unique solution x for all brref (A) = Inrank (A) = nim (A) = Rnker (A) = 0The column vectors of A form a basis of RnThe column vectors of A span RnThe column vectors of A are linearly independentdet A ≠ 00 fails to be an eigenvalue of A
9 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 11 0
10 Example B answer Permutation Matrix 0 11 0Permutation MatrixWhat 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 valuesWhat is a vector with λ = -1? [-1;1] any with opposite values
11 Rotation matrixWhat are the eigenvalues and eigenvectors of a matrix that rotates all vectors 90º?Recall 2x2 rotation matrices have the form:
12 Rotation matrixThere 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.
13 How can I solve Ax = λx Bring everything on one side Ax – λx = 0 (A- λI)x = 0If this can be solved then the matrix(A- λI) must be singularWhich means that det (A- λI) = 0This 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 = 0to find the x’s (Eigenvectors)
15 Find the Eigenvalues Find det (A- λI) = 0 Plug in 3 11 3Find the Eigenvalues1 11 1Note: this equation iscalled the characteristicequationFind det (A- λI) = Plug in(3- λ)2 – λ = 2 and findλ2 - 6 λ + 8 = a basis for kernel(λ-4) (λ-2) = 0λ = 4 and λ = 2Plug in λ = 4 to find the Eigenvectorsfind a basis for the null space (kernel)3- λ 1λ-1 1
16 Eigenvalues of triangular matrices Find the Eigenvalues of3 10 3
17 Triangular matrices slide 1 of solutions 3 10 3Find the EigenvaluesA- λI =det (A) = (3 – λ)2 = λ = 3This matrix has a repeated Eigenvalue.Note: for triangular matrices, the values on the diagonal of the matrix are the Eigenvalues.A=3- λ 1λ
18 Triangular matrices Find the Eigenvectors A- λI = Replace λ by 3 Find the null spaceThis matrix has only 1 Eigenvector!A repeated λ gives the possibility of a lack of Eigenvectors3- λ 1λ
19 Facts about Eigenvalues 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.
21 Eigenvalues and Eigenvectors on the TI89 Calculator 1 23 4Find the eigenvalues and eigenvectors on the calculators2nd 5 (math)4 (matrix)8 (eigVl)eigvl([1,2;3,4])2nd 5 (math)4 (matrix)9 (eigVc)eigVc([1,2;3,4])
22 Homework: worksheet 7.1 5-10 all textbook p.305 15-21 all