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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: definition and some characterizations/results Householder Transforms : basic computation Unitary Equivalence: definition and some conditions Clarification If A M n ( C ) , then: Ax, x = 0 x C n i ff A = 0 . However, in the case where F = R , we require an extra condition: If A M n ( R ) and A = A , then: Ax, x = 0 x R n i ff A = 0 . Warm-up Suppose A M n has orthogonal columns, all of which are non-zero. Can we compute A 1 easily? From 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 = 0 . 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 0 0 . . . 0 0 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 , with eigenvalues λ 1 , λ 2 , ..., λ n (where multiplicity is counted). Then there exists U M n , where U U = I , such that: A = UTU and T = λ 1 · · · 0 λ 2 · · · 0 0 . . . 0 0 · · · 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 , λ 2 , ..., λ n . Choose an eigenvalue x corresponding to λ 1 and, WLOG, sup- pose || x || = 1 . Now extend
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