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Unformatted text preview: CS205 Homework #3 Problem 1 Consider an n n matrix A . 1. Show that if A has distinct eigenvalues all the corresponding eigenvectors are linearly independent. 2. Show that if A has a full set of eigenvectors (i.e. any eigenvalue with multiplicity k has k corresponding linearly independent eigenvectors), it can be written as A = QQ- 1 where is a diagonal matrix of A s eigenvalues and Q s columns are A s eigenvectors. Hint: show that AQ = Q and that Q is invertible. 3. If A is symmetric show that any two eigenvectors corresponding to different eigenvalues are orthogonal. 4. If A is symmetric show that it has a full set of eigenvectors. Hint: If ( , q ) is an eigenvalue, eigenvector ( q normalized) pair and is of multiplicity k > 1, show that A- qq T has an eigenvalue of with multiplicity k- 1. To show that consider the Householder matrix H such that Hq = e 1 and note that HAH- 1 = HAH and A are similar....
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- Fall '07