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Unformatted text preview: Homework 3 1 Basics 1. Find the solutions to the equation 1 2 2 4 x y = 3 6 We notice that the equation x + 2 y = 3 is equivalent to the equation 2 x + 4 y = 6. Hence, all points on the line x + 2 y = 3 are solutions to the above equation, or x y = 1 1 2 + 3 2 2. Matlab gives me a warning that the matrix A is singular. This is because the matrix we give them is noninvertible and hence we cant solve the linear system of equations uniquely. 2 SPD matrix properties 1. Show that the diagonal elements of A are all positive. Because A is symmetric and positive definite, we know that there exists a triangular matrix B such that A = B T B . Now by definition of A being positive definite, we also know that x T Ax > x 6 = 0. The LHS of this second condition can be expanded using the A = B T B property into x T B T Bx = ( Bx ) T Bx > x 6 = 0. Now consider some B of dimension n n , we know that the diagonal of A = B T B must be a sum of squares of the i th column of B . The following example illustrates the 2...
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 Spring '08
 KLEINBERG
 Algorithms

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