MidTerm2005

In addition we have the singular value decomposition

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Unformatted text preview: full rank. In addition, we have the singular value decomposition of B given as: B = U 11V H 1 a* What is the dimension of the nullspace of B ? 7 8 b* Show that by using singular value decomposition from 8, the cost function kBwk2 2 given in the optimization problem from 7 can be expressed as vH Dv, where v 2 C N is a unit norm vector and D 2 C N N is a diagonal square matrix. Express v and D in terms of U 1, 1 and V 1. Name: Matr.-Nr.: Let us denote in the following the singular values of B as s1 s2    sN . c Show that vHDv  s2 N 8 9 Hint: aH Tc = Pi aTi ci , where T is a diagonal matrix and ai is the ith element of the i vector a, Ti is the ith element along the diagonal of the matrix T , and ci is the ith element of the vector c. Name: Matr.-Nr.: d* For which vector v is the expression given in 9 an equality? e* What is the solution w given in terms of V 1 to the optimization problem given in 7? 9...
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