hw7 - Q R where R = p R 11 R 12 P and R 11 is an invertible...

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ECE210A November 7, 2010 Homework 7 Due date: November 14, 2010 1. Reading assignment . Read the proof of the Jordan decomposition theorem in the class notes and supply all the missing details. (No need to turn this in). 2. Reading assignment . Read the chapter on determinants from Meyer or Strang. You should feel comfortable computing determinants of 3 × 3 matrices and Fnding their eigen- values. 3. Let A be a full column-rank matrix. Let x LS = ( A T A ) - 1 A T b . Prove that b A x LS b b 2 = min x b Ax b b 2 . 4. Let A be a full row-rank matrx. Let x MN = A T ( AA T ) - 1 b . Prove that b x MN b 2 = min Ax = b b x b 2 . 5. Let A be an m × n real matrix with rank r . Show that there exists a permutation matrix Π and an orthogonal matrix Q such that A Π =
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Unformatted text preview: Q R, where R = p R 11 R 12 P , and R 11 is an invertible r × r upper-triangular matrix. Also show that the Frst r columns of Q form a basis for the column-space of A . In what fundamental sub-space of A do the last m − r columns of Q lie? Do they form a basis for that space? 6. Assuming A has full column-rank, show how to use the Q R factorization of A from problem 5 to compute the least-squares solution min x b A x − b b 2 . 7. Assuming A has full row-rank, show how to use the Q R factorization of A T from problem 5 to Fnd the minimum-norm solution min Ax = b b x b 2 . 1...
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