Scientific Computing

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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 λ with multiplicity k has k corresponding linearly independent eigenvectors), it can be written as A = QΛQ - 1 where Λ is a diagonal matrix of A ’s eigenvalues and Q ’s columns are A ’s eigenvectors. Hint: show that AQ = 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 ( λ , q ) is an eigenvalue, eigenvector ( q normalized) pair and λ is of multiplicity k > 1, show that A - λ qq T has an eigenvalue of λ 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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