Scientific Computing

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CS205 Homework #7 Problem 1 We have seen the application of the conjugate gradient algorithm on the solution of sym- metric, positive definite systems. Now assume that in the system Ax = b , the n × n matrix A is symmetric positive semi-definite with a nullspace of dimension p < n . This problem illustrates that one can use a modified version of conjugate gradients to solve this system as well. 1. Prove that we can write A as A = M ˜ AM T where M is an n × ( n - p ) matrix with orthonormal columns that form a basis for the column space of A , while ˜ A is an ( n - p ) × ( n - p ) symmetric positive definite matrix (no nullspace) [Hint: Use the diagonal form of A = QΛQ T ] 2. Let the n × n matrix P be defined as P = MM T . Explain (no formal proof required) why this is a projection matrix and onto what space it projects. How can we compute P without knowledge of the eigenvalues-eigenvectors of A ? 3. Show that, in order to have a solution to Ax = b , we must be able to write b = M ˜ b for an appropriate vector ˜ b R n - p 4. Let ˜x
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