4 Eigenvalues and Eigenvectors - Solutions

4. Eigenvalues and Eigenvectors (References: Comps Study Guide for Linear Algebra Section 4; Damiano & Little, A Course in Linear Algebra, Chapter 4) Let A be an n ⇥ n matrix, 2 R be a scalar, and let ~v 2 R. To say ~v is an eigenvector of A with eigenvalue means ~v 6= 0 and A~v = ~v . Eigenvalues: Write down what it means to say that is an eigenvalue of A. Eigenvector: Write down what it means to say that ~v is an eigenvector of A. Ex. 1. Let A be an n ⇥ n matrix. Prove that A is invertible if and only if = 0 is not an eigenvalue of A. (Note: This is a common theorem that you could usually use without proof.) Ex. 2. Suppose that A is an invertible n ⇥ n matrix. Prove that if ~x is an eigenvector of A with eigenvalue 1 then ~x is also an eigenvector of A 1 with eigenvalue . Characteristic Polynomial & Finding Eigenvalues: The characteristic polynomial of A is det(A eigenvalues of A are the roots of the characteristic polynomial. 2 2 Ex. 3. Find the eigenvalues of the matrix A = 40 1 1 1 1 I). The 3 2 0 5. 1 Eigenspaces & Finding Eigenvectors: The eigenspace E of an eigenvalue is the nullspace N (A the matrix A I. The eigenvectors of A with eigenvalue are the nonzero elements of E . 2 3 2 1 2 Ex. 4. For a basis for the eigenspace of each eigenvalue of the matrix A = 40 1 0 5. 1 1 1 I) of Algebraic Multiplicity: The algebraic multiplicity of characteristic polynomial. is the number of times it appears as a root of the Geometric Multiplicity: The geometric multiplicity of is the dimension dim(E ) of the eigenspace E . Theorem: For any eigenvalue , 1  (geometric multiplicity of )  (algebraic multiplicity of ). Diagonalizability: Write down what it means to say that an n ⇥ n matrix A is diagonalizable. Theorem: Let A be an n ⇥ n matrix. The following are equivalent. (1) A is diagonalizable. (2) There is an invertible matrix P and a diagonal matrix D such that P 1 AP = D. (3) There is a basis of Rn consisting of eigenvectors of A (i.e., an eigenbasis). (4) The characteristic polynomial has n real roots (possibly repeated) and for each root , (geometric multiplicity of ) = (algebraic multiplicity of ). In this case, the basis of eigenvectors are the columns of P and the corresponding eigenvalues are the diagonal entries in the corresponding columns of D (i.e., in the same order). Remark: This comes from the fact that if T is the linear map T (x) = Ax and ↵ is an eigenbasis, then std D = [T ]↵ ↵ is the matrix of T with respect to ↵ and P = [I]↵ is the change of basis matrix from the basis ↵ n to standard coordinates on R . Note: It follows that if A has n distinct real eigenvalues, then A is diagonalizable. However, if A has repeated eigenvalues, it may or may not be diagonalizable. Ex. 5. Determine whether or not the matrix A from examples 3 and 4 is diagonalizable. If it is, find an invertible matrix P and a diagonal matrix D such that P 1 AP = D. 2 3 2 1 2 A = 40 1 0 5 1 1 1 . Ex. 6. Let the matrix A be as defined below. Find a diagonal matrix D and an invertible matrix P such that A = P DP 1 , or show that no such matrices exist. 2 3 0 4 2 2 15 . A = 40 0 1 0 Additional Problems 2 3 2 0 1 55. Ex. 7. Consider the matrix A = 43 0 1 0 0 (a) Find all eigenvalues of A. (b) Find a diagonal matrix D and an invertible matrix P such that A = P DP matrices exist. 2 3 2 1 1 2 0 5. Ex. 8. Let A be the matrix A = 4 0 6 1 3 (a) Find all eigenvalues of A. (b) Find a a basis of each eigenspace. (c) Find a diagonal matrix D and an invertible matrix P such that D = P AP matrices exist. Ex. 9. Consider the matrix 2 1 A=4 0 2 Find a basis for R3 consisting of eigenvectors of A, or Ex. 10. Consider the matrix Is A diagonalizable? Why or why not? 2 1 A = 42 1 1 , or show that no such 1 , or show that no such 0 0 0 else 3 2 0 5. 4 prove that there is no such basis. 2 5 2 3 0 05 . 2 Ex. 11. Let V be a finite-dimensional vector space, and let T : V ! V be a linear transformation. Prove that 0 is an eigenvalue of T if and only if the image (i.e., range) of T is not equal to V . Ex. 12. Suppose that a 3 ⇥ 3 matrix A has 0 as an eigenvalue. (a) What are the possible values of the rank of A? Justify your answer. (b) Let T : R3 ! R3 be the linear transformation given by T (x) = Ax. Can T possibly be one-to-one? Can T be onto? Justify your answer. Ex. 13. Let A, B be n ⇥ n matrices that commute, i.e. AB = BA. Let v 2 Rn be an eigenvector of A such that Bv = 6 0. Prove that Bv is also an eigenvector of A. Ex. 14. Prove that if T : V ! V is a linear map then the eigenspace E0 corresponding to the eigenvalue = 0 is equal to Ker(T ). Ex. 15. Suppose that A is an n ⇥ n matrix that satisfies A2 = I, where I is the n ⇥ n identity matrix. Show that if is an eigenvalue of A then = 1 or = 1. Ex. 16. Let A be an n ⇥ n matrix and all integers m 1. be an eigenvalue of A. Prove that m is an eigenvalue of Am for Ex. 17. Let A be an n ⇥ n matrix and ↵ 2 R be a scalar that is NOT an eigenvalue of A. Suppose that µ is an eigenvalue for the matrix B = (A ↵I) 1 with corresponding eigenvector v. Prove that v is also an eigenvector for A and find a formula for the corresponding eigenvalue of A in terms of µ and ↵.