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midterm-1 - Spring 2010 CS530 Analysis of Algorithms...

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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 don’t know”, but if you write something that is wrong, you may get less. (It is not possible to pass just writing “I don’t 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. Solution. We have to show x T C x 0. This can be written as ( B T x ) T A ( B T x ) = y T Ay with y = B T x . The semidefiniteness of A
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