Lecture #7 Notes

Then t t u au u au u au u a u u aa u u

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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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