Linear Algebra Review Session 4 Eigenvalues and Eigenvectors

Linear Algebra: Review Session 4 Eigenvalues and Eigenvectors Computations and Diagonalizability   2 1 −2 Ex. 1. Let A be the matrix A = 0 1 0 . 1 1 −1 (a) Define what it means for a real number λ ∈ R to be an eigenvalue of A. (b) Find all eigenvectors of A. (c) Find a diagonal matrix D and an invertible matrix P such that A = P DP −1 , or show that no such matrices exist.   2 0 −1 Ex. 2. Consider the matrix A = 3 0 −5. 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 −1 , or show that no such matrices exist. Ex. 3. Let A be the matrix  2 1 1 2 0 . A= 0 −6 −1 −3 Find a diagonal matrix D and an invertible matrix P such that D = P AP −1 , or show that no such matrices exist. Ex. 4. Consider the matrix   1 A= 0 −2 3 Find a basis for R consisting of eigenvectors of A, or Ex. 5. Consider the matrix  1 A = 2 1  0 −2 0 0 . 0 4 else prove that there is no such basis. −2 5 2  0 0 . 2 Is A diagonalizable? Why or why not? Proofs Ex. 6. 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. 7. Let A, B be n × n matrices that commute, i.e. AB = BA. Let v ∈ Rn be an eigenvector of A such that Bv = 6 0. Prove that Bv is also an eigenvector of A. Ex. 8. 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. 9. Let A be an n × n matrix and α ∈ R be a scalar that is NOT an eigenvalue of A. Suppose that µ is an eigenvalue for the matrix B = (A − αI)−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 α. Ex. 10. Prove that if T : V → V is a linear map then the eigenspace E0 corresponding to the eigenvalue λ = 0 is equal to Ker(T ).