midterm1 fall07

Scientific Computing

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CS 205A Fall 2007 Midterm Multiple Choice (8 x 1 pt) 1. For a symmetric ( A = A T ) n × n matrix A , (a) A T A is invertible (b) The eigenvalues are all real-valued (c) For any b , there exists an x such that Ax = b (d) There exist non-singular matrices L and U such that A = LU 2. If we’re given a matrix A , the following can be said: (a) If A is positive definite, then it has Cholesky Factors (ie A = LL T ) (b) If A is over-determined, we use QR decomposition with Householder to solve Ax = b (c) The pseudo-inverse of A , A + = V Σ + U T can be used to solve least-squares problems (d) Solving the minimization problem min x b - Ax 2 is equivalent to solving Ax = b 3. The following can be said about a Householder Matrix H = I - 2 vv T v T v (a) The condition number for H is (b) It is a projection matrix onto the hyperplane orthogonal to v (c) It preserves the 2-norm of a vector (ie x 2 = Hx 2 for all x ) (d) The eigenvalues of H are 1 with multiplicity n 4. Consider the multi-variate optimization problem min x f ( x ).
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