Lecture #7 Notes - Matrix Theory Math6304 Lecture Notes...

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Matrix Theory, Math6304 Lecture Notes from September 18, 2012 taken by John Haas Last Time (9/13/12) Unitary Matrices: defnition and some characterizations/results Householder Transforms :ba s i ccompu ta t ion Unitary Equivalence: defnition and some conditions ClariFcation IF A M n ( C ) ,then : ° Ax, x ± =0 x C n A . However, in the case where F = R ,werequ ireanextracond it : IF A M n ( R ) and A = A : ° Ax, x ± x R n A . Warm-up Suppose A M n has orthogonal columns, all oF which are non-zero. Can we compute A 1 easily? ²rom our characterizations oF unitary matrices From last time, we know we can write: [ A ] ij = d i u ij ,where U =( u ij ) is unitary and d i ² . Hence, by using our knowledge oF right multiplication with diagonal matrices, we get: A=UD, where d i is the ith entry along the the diagonal oF the diagonal matrix D. So it Follows that: A 1 = D 1 U 1 = D 1 U = 1 d 1 0 ··· 0 0 1 d 2 0 00 . . . 0 1 d n U 1
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Thus, using our knowledge of left multiplication with diagonal matrices, we get: [ A 1 ] ij = d 1 i [ U ] ij = d 1 i ¯ u ji 2 Schur Triangularization and Consequesnces (continued) 2.1 Schur’s triangularization theorem 2.1.1 Theorem. (Schur’s Triangularization Lemma) Let A M n ,w i the igenva lue s λ 1 2 ,...,λ n (where multiplicity is counted). Then there exists U M n ,where U U = I ,suchthat : A = UTU and T = λ 1 ∗· · · ∗ 0 λ 2 ··· 00 . . . λ n is upper triangular. In other words, every square matrix is unitarily equivalent to an upper triangular matrix. Proof. The proof follows by induction on the dimension, n: ( n =1): Done. ( n 1 n ): Suppose Schur’s theorem holds for all ( n 1) × ( n 1) matrices, and let A M n , with eigenvalues λ 1 2 n .C h o o s ea ne i g e n v a l u e x corresponding to λ 1 and, WLOG, sup- pose || x || =1 .Nowe x t end { x } to a basis for C n ,andthenapp lytheG ram -Schm idttogetan
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Lecture #7 Notes - Matrix Theory Math6304 Lecture Notes...

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