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Unformatted text preview: an equate:
tn,n 2 = t1,n 2 + t2,n 2 + ... + tn,n 2 ,
which means:
n −1
tk,n 2 = 0.
k=1 Hence, T is of the form: T = T 0
0
.
.
. 0 · · · 0 tn,n . Now T ∈ Mn−1 , and, since T is upper triangular and normal, it follows that T must also be
upper triangular and normal, so we may now invoke induction assumption. That is to say, T
must be diagonal, and, hence, T is diagonal.
2.2.7 Theorem. (Spectral Theorem) Let A ∈ Mn be normal. Then A is unitarily equivalent to
a diagonal matrix.
Proof. Let A ∈ Mn be normal, ie, AA∗ = A∗ A. Then it follows from Schur’s theorem that there
exists a unitary U ∈ Mn and an upper triangular T ∈ Mn such that T = U ∗ AU . Then:
T T ∗ = (U ∗ AU )(U ∗ AU )∗...
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 Fall '12
 BernhardBodmann
 Matrices

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