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# GSW Problem Set 5 Answers - Engineering Analysis 4 Workshop...

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Engineering Analysis 4 Workshop questions - week of 11/09/08 1. Give an example of a matrix which has repeated eigenvalues, but which is not defective. Answer - Defective matrices are characterized by having repeated eigenvalues, but not enough eigenvectors for each of the eigenvalues. A non-defective matrix can have a repeated eigen- value, but it must have two linearly independent eigenvectors to go along with the eigenvalue. A possible matrix is ± 4 0 0 4 ² , which has 4 as a repeated eigenvalue. It is easy to see that this matrix has two linearly inde- pendent eigenvectors, for example, ~ v 1 = ± 1 0 ² , ~ v 2 = ± 0 1 ² . 2. Consider the forced system ~x 0 = A~x + te 2 t ~ f , (1) where ~ f a constant vector and A is a matrix. (a) Set up the equations to Fnd a particular solution using undetermined coefFcient vectors. Answer - You have to take as trial functions the forcing term and all of its derivatives. Thus, we try

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GSW Problem Set 5 Answers - Engineering Analysis 4 Workshop...

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