NumericalAnalysis-682-2008aug - A 2 is an arbitrary matrix...

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MAT 682 Numerical Analysis August, 2008 NAME: 1. Suppose A be an n × n invertible matrix. Let A = U Σ V * be the singular value decomposition of A , where Σ = diag { σ 1 2 ,...,σ n } and σ 1 σ 2 ≥ ··· ≥ σ n > 0 . (a) Show that κ 2 ( A ) = σ 1 n , where κ 2 ( A ) ≡ k A k 2 k A - 1 k 2 . (b) Show that κ 2 ( A ) ( n - 1) k A k 2 k A k 2 F - k A k 2 2 (Hint: Show k A k 2 F σ 2 1 + ( n - 1) σ 2 n ) (c) Show that the above inequality is sharp, by constructing a 22 × 22 invertible matrix with k A k 2 = 10 and k A k F = 11 such that κ 2 ( A ) = 10. 2. (a) Show that every 2 × 2 Householder matrix (reflector) H is of the form H = a b b - a where a 2 + b 2 = 1. (b) Find a Householder matrix H that annihilates the second component of the vector x = 4 3 . Also find Hx . 3. (a) Let A R m × n ( m > n ). For x R m , let x 1 Px where P is the orthogonal projection matrix that projects R m onto R ( A ) (range of A ). Show that A T ( x - x 1 ) = 0 for all x R m and k x k 2 ≤ k x 1 k 2 . (b) Let the matrix A be of the form A = A 1 A 2 , where A 1 is non-singular of dimension n × n and
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Unformatted text preview: A 2 is an arbitrary matrix of dimension ( m-n ) n . Show that k A + k 2 k A-1 1 k 2 , where A + ( A T A )-1 A T . 4. Let A be an n n real symmetric positive denite symmetric matrix of the form A = a 11 v T v K where v R n-1 . (a) Show that a 11 > 0. 1 (b) Let = a 11 and consider the factorization A = v/ I n-1 1 K-v v T /a 11 v T / I n-1 Show that the matrix K-v v T /a 11 is symmetric and positive denite. (c) Let B be an n n real non-singular matrix. Suppose B = QR is the QR-decomposition of B and suppose B T B = U T U is the Cholesky decomposition of B T B . Show that R = U . 2...
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This note was uploaded on 06/19/2011 for the course MATH 684 taught by Professor Na during the Spring '11 term at University of North Carolina School of the Arts.

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NumericalAnalysis-682-2008aug - A 2 is an arbitrary matrix...

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