e483h3_07

# e483h3_07 - Simon Fraser University School of Engineering...

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School of Engineering Science ENSC 483 - Modern Control Systems Spring 2007–Assignment 3 1. (a) Find the eigenvalues of the following matrices: A = 8 - 8 - 2 4 - 3 - 2 3 - 4 1 ; B = 2 1 1 0 3 1 0 - 1 1 ; C = α ω - ω α ; D = 0 1 - ω 2 n 0 (b) Discuss if the above matrices can be diagnonilized. 2. Consider the following symmetric matrix A = 3 0 - 2 0 2 0 - 2 0 0 (a) Find the eigenvalues and the eigenvectors of this matrix. (b) Find the (special) modal matrix Q whose columns are eigenvectors normalized to be of unit length. (c) Form the product ˆ A = Q T AQ and show that ˆ A is a diagonal matrix. (d) What do you conclude from this exercise? 3. Show that if an eigenvalue of an n × n matrix A is zero, then A is singular. 4. (a) Let T be any non-singular n × n matrix such that Tv 1 = e 1 , where e 1 is the ﬁrst column of I n , and v 1 is an eigenvector associated with the eigenvalue λ 1 of an n × n matrix A . Show that TAT - 1 = λ 1 . . . b ··· ··· ··· 0 . . . ˜ A where elements of b are unimportant. Hence deduce that the eigenvalues of the ( n - 1) × ( n - 1) matrix ˜ A are the remaining ( n - 1) eigenvalues of

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e483h3_07 - Simon Fraser University School of Engineering...

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