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quiz2solutions

# quiz2solutions - MIT OpenCourseWare http/ocw.mit.edu 18.085...

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MIT OpenCourseWare http://ocw.mit.edu 18.085 Computational Science and Engineering I Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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18.085 Quiz 2 November 2, 2007 Professor Strang Your PRINTED name is: Grading 1 2 3 1) (40 pts.) This problem is based on a 5-node graph. I have not included edge numbers and arrows. Add them if you want to: not needed. (a) Find A T A for this graph. A is the incidence matrix. (b) The sum of the eigenvalues of A T A is . The product of those eigenvalues is . (c) What is A T A for a graph with only one edge ? How can that small A T A be used in constructing A T A for a large graph ? (d) Suppose I want to solve Au = ones(8 , 1) = b by least squares. What equation gives a best u ? For the incidence matrix A , is there exactly one best u solving that equation ? (If your equation has more than one best u , describe the difference between any two solutions.) 1 2 3 4 5 1
c a The matrix A t A is the graph Laplacian, given by D W were D is the degree matrix and W is the adjacency matrix. Therefore it is a 5 × 5 matrix: 3 1 1 0 1 1 3 0 1 1 1 0 3 1 1 0 1 1 3 1 1 1 1 1 4 b The trace of A t A is the sum of the diagonal entries, so 16. The determinant is the product of the eigenvalues. Since any graph Laplacian has a null-vector (namely any constant vector), one of the eigenvalues is 0, so the determinant is also 0. The A t A for a one edge graph i 1 1 1 1 To construct A t A for a larger graph from these element matrices, we construct one for each edge of the graph and add them together according to the following scheme. If there are n nodes, start with an n × n matrix of zeroes M . Then, for an edge connecting node i to node j , add 1 to M ii and

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quiz2solutions - MIT OpenCourseWare http/ocw.mit.edu 18.085...

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