Lecture #7 Notes

# So it follows that a1 d1 u 1 d 1 u 1 0 d1 0 1 d2

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Unformatted text preview: entry along the the diagonal of the diagonal matrix D. So it follows that: A−1 = D−1 U −1 = D −1 U ∗ 1 0 ··· d1 0 1 ··· d2 = 0 0 ... 0 0 ··· 1 0 0 ∗ U 0 1 dn Thus, using our knowledge of left multiplication with diagonal matrices, we get: [A−1 ]ij = d−1 [U ∗ ]ij = d−1 uji ¯ i i 2 2.1 Schur Triangularization and Consequesnces (continued) Schur’s triangularization theorem 2.1.1 Theorem. (Schur’s Triangularization Lemma) Let A ∈ Mn , with eigenvalues λ1 , λ2 , ..., λn (where multiplicity is counted). Then there exists U ∈ Mn , where U ∗ U = I , such that: A = UT U∗ and λ1 ∗ · · · ∗ 0 λ2 · · · ∗ T = 0 0 ... ∗ 0 0 · · · λn is upper triangular. In other words, every sq...
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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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