Scientific Computing

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CS205 Review Session #7 Notes Useful Decompositions Consider an n × n matrix A that is symmetric positive semi-definite and has a nullspace of dimension p < n . We wish to factor A into M ˜ AM where the columns of the n × ( p - n ) matrix M form an orthonormal basis for col( A ) = row( A ). It may help to first consider a concrete example: A = 2 - 1 0 - 1 2 0 0 0 0 = 1 0 0 1 0 0 2 - 1 - 1 2 1 0 0 0 1 0 = M ˜ AM T This particular choice for ˜ A and M is not necessarily unique. We can also write: A = 2 - 1 0 - 1 2 0 0 0 0 = 1 2 - 1 2 1 2 1 2 0 0 1 0 0 3 1 2 1 2 0 - 1 2 1 2 0 One way to construct this factorization in general is to compute the spectral decomposi- tion A = QΛQ T of A and trim of some of the unnecessary columns of Q . We can show, however, that no matter what method we use to determine A = M ˜ AM T , if the columns of M form an orthonormal basis for the column space of A , ˜ A must have several desirable properties. In particular: 1.
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