1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Suppose J(A) R5x5 is a Jordan block of some eigenvalue A. Find J* (A). Suppose ai is the (i, j)th entry of the matrix A of order n, where Find the characteristic polynomial of A, or AI - A. Find limno A where -G-1 1 0 Let x== x= [1 2 3]. In the following two cases, decide respectively if x Ax 0 for each x: a). b). A = [71] P.q (0, 1). 1-q A = A = c). 3 -2 4 5 1 4 -1 if i=n, if j = i + 1, otherwise. Suppose A E R* and BE Rxm and A 0. Show that A is an eigenvalue of AB if and only if A is an eigenvalue of BA. Let an = 3a-1 + 2an-2 for n = 2,3,... Find Hint: Different initial values of ao, an lead to different answers. Suppose A, A2,,As are the eigenvalues of AT A where A ^= [ 1 A = A = ak lim k-x ak-1 1 2 -3 0 1 01 1 -2 -1 Find A. Find II, A. c). Find . . Find a second order polynomial f(z) that best fits the points Find the eigevalues of vv. Find the rank of vv. Show that viv is not diagonalizable. 1 2 1 242 12 (, )= (-1, 1), (12. 2) = (0,2), (3, 33) = (1,-1), (4, 34) = (2, 1). Namely, with this polynomial the error (f(x)- y) is minimal. Using the algebraic multiplicity, geometric multiplicity, and the index of eigenvalues to give two conditions under either one a square matrix is diagonalizable. Suppose nonzero vectors V, V R and v is orthogonal to v. a). b).