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Unformatted text preview: Spring 2010 CS530 Analysis of Algorithms Midterm exam 1 Midterm exam 1 Only a single hand-written crib sheet can be used, no books or notes. Even if I ask for just a yes / no answer, you must always give a proof. You may get some points even if you write I dont know, but if you write something that is wrong, you may get less. (It is not possible to pass just writing I dont know everywhere. . . .) Problem 1. (10pts) Show that the inverse of a permutation matrix is its transpose. Solution. Let e i be the standard basis vector that has a 1 at position i and 0 elsewhere. Then if P = ( p i j ) is a permutation matrix and p i j = 1 then Pe j = e i . This is equivalent to P- 1 e i = e j , which shows that P- 1 is the transpose of P . Problem 2. Consider an n n a symmetric matrix A and an n n matrix B . Let C = BAB T . (a) (5pts) Show that if A is positive semidefinite then C is also....
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- Spring '09