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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 bottomright entry of T T ∗ :
¯
[T ∗ T ]n,n = tn,n tn,n = tn,n 2
Next, look at the bottomright 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.
 Fall '12
 BernhardBodmann
 Matrices

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