Lecture#7 Notes

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Unformatted text preview: = U ∗ AU U ∗ A∗ U = U ∗ AA∗ U = U ∗ A∗ AU = U ∗ A∗ U U ∗ AU = (U ∗ AU )∗ (U ∗ AU ) = T ∗T Thus, T is normal and upper triangular, so it follows from the previous proposition that T must be diagonal. We can strengthen this theorem. 2.2.8 Theorem. (”Super” Spectral Theorem) Let A ∈ Mn . Then the following are equivalent: (i) A is normal. (ii) A is unitarily diagonalizable. n n ￿ ￿ (iii) | λj | 2 = |ai,j |2 , where λ1 , λ2 , ..., λn are the eigenvalues of A (multiply counted). j =1 i,j =1 Proof. (i) ⇒ (ii) 5 Shown in the previous theorem. (ii) ⇒ (i) Suppose A = U DU ∗ for some diagonal D ∈ Mn and unitary U ∈ Mn . Then: A∗ A = (U DU ∗ )∗ (U DU...
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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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