Theorem 6.14 (Schur). Let T be a linear operator on a nite- dimensional inner product space V. Suppose that the characteristic poly- nomial of T splits. Then there exists an orthonormal basis 7 for V such that the matrix [T]? is upper triangular. Proof. By Exercise 12(a) of Section 5.2, there exists an ordered basis [5' = {w1,w2, . . . ,wn} for V such that [T] 5 is upper triangular. New apply the Gram-Schmidt process to 6 to obtain an orthogonal basis 5' = {v1,v2, . . . .1271} for V. For each It, 1 g k g 77., let 5k 2 {w1,w2,...,wk} and S}; = {$11,112, . . . ,vk}. As in the proof of Theorem 6.4, span(Sk) = span(S,;) for all It. By Exercise 12 of Section 2.2, T(wk) E span(8k) for all k. Hence T(vk) E span(S};) for all k, and so [That is upper triangular by the same exercise. Finally, let 2.- = u; for all 1 g i g n and \"y = {Z17 22, . . . , 2\"}. Then 7 is an orthonormal basis for V, and [T]., is upper triangular. I 12. (a) Prove that if A E Mnxn(F) and the characteristic polynomial of A splits, then A is similar to an upper triangular matrix. (This proves the converse of Exercise 9(b).) Hint: Use mathematical induction on n. For the general case, let '01 be an eigenvector of A, and extend {'01} to a basis {U1,'02,...,'0n} for F\". Let P be the n X 77. matrix Whose jth column is '03-, and consider PlAP. Exercise 13(a) in Section 5.1 and Exercise 21 in Section 4.3 can be helpful. (b) Prove the converse of Exercise 9(a). Visit goo . gl/gJSjRU for a solution. Let T be a linear operator on a nite-dimensional vector space V, and suppose there exists an ordered basis [3 for V such that [T] g is an upper triangular matrix. (a) Prove that the characteristic polynomial for T splits