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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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- Fall '06
- Matrices, Diagonal matrix, Golub, Van Loan Problem