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# hw2 - LU factorizations of A and B When are there no...

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1 Home Work 2 Due on October 11, 2010 Reading assignments don’t have to be turned in. 1. Reading assignment. Finish reading chapter 2 of the notes posted on the class web-site. 2. Reading assignment. Read chapter 3 of the notes posted on the class web-site. 3. Suppose A R n × n , b R n , and that φ ( x ) = 1 2 x T Ax x T b . Show that the gradient of φ is given by φ ( x ) = 1 2 ( A T + A ) x b . 4. Let A = ( 1 / 3 1 / 3 1 / 3 ) T . Think of A as an operator from R 1 to R 3 via matrix-vector multiplication. Show that the operator is one-to-one. Find two linear left-inverses for A . Find a left-inverse for A that is not linear. 5. Find all matrices X that satisfy the equation AXB T = C, in terms of the
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Unformatted text preview: LU factorizations of A and B . When are there no solutions? 6. Let U 1 and U 2 be two upper-triangular matrices. Let Z be an m × n matrix. Let X be an unknown matrix that satisﬁes the equation U 1 X + XU 2 = Z. A. Give an algorithm to ﬁnd X in O ( mn ( m + n )) ﬂops (ﬂoating-point opera-tions). B. Find conditions on U 1 and U 2 which guarantee the existence of a unique solution X . C. Give a non-trivial example ( U 1 6 = 0, U 2 6 = 0, X 6 = 0) where those conditions are not satisﬁed and U 1 X + XU 2 = 0 ....
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