tnn 2 k1 k1 4 by the normality of t we get t

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Unformatted text preview: repare for our diagonalizability theorem, we examine the behaviors of normal matrices vs triangular matrices. 2.2.6 Proposition. Let T ∈ Mn be triangular. Then T is normal iff T is diagonal. Proof. (⇐) : Trivial. (⇒) : We proceed by induction on the dimension, n: (n = 1) : Done. (n − 1 ⇒ n) : Suppose this proposition holds for all A ∈ Mn−1 , and consider the triangular matrix T where: t1,n t2,n T = . , . . 0 · · · 0 tn,n T￿ Now, look at the bottom-right entry of T T ∗ : ¯ [T ∗ T ]n,n = tn,n tn,n = |tn,n |2 Next, look at the bottom-right entry of T ∗ T : n n ￿ ￿ ¯ [T ∗ T ]n,n = tk,n tk,n = |tk,n |2 = |t1,n |2 + |t2,n |2 + ... + |tn,n |2 k=1 k=1 4 By the normality of T, we get: [T T ∗ ]n,n = [T ∗ T ]n,n Thus, we c...
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This note was uploaded on 01/16/2014 for the course MATH 6304 taught by Professor Bernhardbodmann during the Fall '12 term at University of Houston.

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