5.1 please only write on paper. no explanation needed. Questions: 28, 33, 35 on paper

300 CHAPTER 5 Eigenvalues and Eigenvectors 26. The eigenvectors that we have been studying are sometimes called right eigenvectors to distinguish them from left eigen- 34. Prove: If A is an eigenvalue of A, x is a corresponding eigen- vectors, which are n x 1 column matrices x that satisfy the vector, and s is a scalar, then A - s is an eigenvalue of A - SI equation x A = ux for some scalar M. For a given matrix A, and x is a corresponding eigenvector. how are the right eigenvectors and their corresponding eigen- values related to the left eigenvectors and their corresponding 35. Prove: If A is an eigenvalue of A and x is a corresponding eigenvalues? eigenvector, then $1 is an eigenvalue of SA for every scalar s and x is a corresponding eigenvector. 27. Find a 3 x 3 matrix A that has eigenvalues 1, -1, and 0, and for which 36. Find the eigenvalues and bases for the eigenspaces of and then use Exercises 33 and 34 to find the eigenvalues and are their corresponding eigenvectors. bases for the eigenspaces of Working with Proofs a. A-I b. A - 31 C. A + 21 28. Prove that the characteristic equation of a 2 x 2 matrix A can 37. Prove that the characteristic polynomial of an n x n matrix A has degree n and that the coefficient of 1" in that polynomial be expressed as 12 - tr(A)A + det(A) = 0, where tr(A) is the trace of A. is 1. 38. a. Prove that if A is a square matrix, then A and AT have the 29. Use the result in Exercise 28 to show that if same eigenvalues. [Hint: Look at the characteristic equa- A = a tion det (1I - A) = 0.] b. Show that A and AT need not have the same eigenspaces. [Hint: Use the result in Exercise 30 to find a 2 x 2 matrix then the solutions of the characteristic equation of A are for which A and A have different eigenspaces.] a = = [(a + d) + v(a-d)2+ 4bc 39. Prove that the intersection of any two distinct eigenspaces of Use this result to show that A has a matrix A is {0}. a. two distinct real eigenvalues if (a - d) + 4bc > 0. True-False Exercises b. two repeated real eigenvalues if (a - d) + 4bc = 0. TF. In parts (a)-(f) determine whether the statement is true or c. complex conjugate eigenvalues if (a - d) + 4bc