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# hw4 - T = QR where R ∈ R n × n is an upper triangular...

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Homework 4 Numerical Linear Algebra 1 Fall 2010 Due date: beginning of class Monday, 10/11/08 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 deﬁcient, and show that the SVD of A is all real and has the form A = U ± S 0 ² 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 A R m × n have rank k with k n , i.e., it may be rank deﬁcient. The pseudoinverse behaves much as the inverse for nonsingular matrices. To see this show the following identities are true (Stewart 73) and comment on the eﬀect of rank deﬁciency on each: 4.2.a . AA A = A 4.2.b . A AA = A 4.2.c . A A = ( A A ) T 4.2.d . AA = ( AA ) T 4.2.e . If A R has orthonormal columns then A = A T . Why is this important for consistency with simpler forms of least squares problems that we have discussed? Problem 4.3 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

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Unformatted text preview: T = QR where R ∈ R n × n is an upper triangular matrix and Q ∈ R n × n is an orthogonal matrix. 1 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 diagonal and the ﬁrst two superdiagonals. (4.3.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. (4.3.b) Show that T + = RQ is a symmetric triagonal matrix. (4.3.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 4.4 Golub and Van Loan Problem 8.3.1. p. 423 Problem 4.5 Golub and Van Loan Problem 8.3.6. p. 424 Problem 4.6 Golub and Van Loan Problem 8.3.8. p. 424 2...
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hw4 - T = QR where R ∈ R n × n is an upper triangular...

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