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hw4 - diagonal and the Frst two superdiagonals(4.2.a Show...

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Homework 4 Numerical Linear Algebra 1 Fall 2011 Solutions will be posted Friday, 10/14/11 Problem 4.1 Given that we know the SVD exists for any complex matrix A C m × n , assume that A R m × n has rank k with k n , i.e., A is real and it may be rank deFcient, and show that the SVD of A is all real and has the form A = U p S 0 P V T = U k Σ k V T k where S R n × n is diagonal with nonnegative entries, U = ( U k U m - k ) , U T U = I m V = ( V k V n - k ) , V T V = I n U k R m × k , and V k R n × k Problem 4.2 Let T R n × n be a symmetric tridiagonal matrix, i.e., e T i Te j = e T j Te i and e T i Te j = 0 if j < i - 1 or j > i + 1. Consider T = QR where R R n × n is an upper triangular matrix and Q R n × n is an orthogonal matrix. Recall, the nonzero structure of R was derived in class and shown to be e T i Re j = 0 if j < i (upper triangular assumption) or if j > i + 2, i.e, nonzeros are restricted to the main

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Unformatted text preview: diagonal and the Frst two superdiagonals. (4.2.a) Show that Q has nonzero structure such that e T i Qe j = 0 if j < i-1, i.e., Q is upper Hessenberg. (4.2.b) Show that T + = RQ is a symmetric triagonal matrix. (4.2.c) Prove the Lemma in the class notes that states that choosing the shift μ = λ , where λ is an eigenvalue of T , results in a reduced T + with known eigenvector and eigenvalue. Problem 4.3 Golub and Van Loan Problem 8.3.1. p. 423 1 Problem 4.4 Golub and Van Loan Problem 8.3.6. p. 424 Problem 4.5 Golub and Van Loan Problem 8.3.8. p. 424 2...
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hw4 - diagonal and the Frst two superdiagonals(4.2.a Show...

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