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hw6 - L is defined as L = D-A where D is a diagonal matrix...

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MATH 343:001 (11814) Applied Linear Algebra Fall 2009 Homework 6 (revised) Due Wednesday, December 2 Turn in answers to the following problems. Section 6.2: 2, 18, 24 Section 6.3: 2, 19 Extra Credit (+10pts): Section 8.4: 12, 13 (these two are optional, but the MATLAB problems below are required). MATLAB PROBLEMS. (1) Compute the adjacency matrix A and Laplacian L for the following graphs and verify that a graph is connected if zero is a simple eigenvalue of the graph Laplacian (that is, list the corresponding eigenvalues for each Laplacian). Verify that all eigenvalues are non-negative. Find an upper bound for all the eigenvalues using Gershgorin’s circle theorem (page 373) and confirm it for these four. (You could verify this using your earlier Gershgorin circles MATLAB script.) The adjacency matrix A has a 1 in the ij entry if there is an edge connecting the i and j nodes and a zero otherwise (the matrix is symmetric, it is an undirected graph). The Laplacian
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Unformatted text preview: L is defined as L = D-A where D is a diagonal matrix with d ii equal to the sum of the i th row of A . For example, if A = 0 1 1 0 1 0 1 1 1 1 0 0 0 1 0 0 then D = 2 0 0 0 0 3 0 0 0 0 2 0 0 0 0 1 and L = 2-1-1-1 3-1-1-1-1 2-1 1 . 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 (a) (b) (c) (d) (2) Use the svdcomp2.m MATLAB script posted on the course webpage to display the ‘books.jpg’ image (also online) and its compressed version to at most 10% of original size but still of good enough quality to read the titles. Indicate on the plot title the compression size ( r 2 c /r 2 o where r c , r o are the compressed and original ranks of the images). Gerardo Lafferriere, November 25, 2009...
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