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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.
 Fall '12
 BernhardBodmann
 Matrices

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