Lecture #7 Notes

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 )∗...
View Full Document

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.

Ask a homework question - tutors are online