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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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