e483h3s_07 - 1 ENSC 483 HW#3 Solution–M Saif Simon Fraser...

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ENSC 483 HW#3 Solution–M. Saif 1 Simon Fraser University School of Engineering Science ENSC 483 - Modern Control Systems 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. Solution: (a) In the case of matrix A , the eigenvalues are given by λ 1 = 1 2 = 3 3 = 2 . For matrix B , all the eigenvalues are located at 2 . For matrix C , we have Δ( λ ) = ( α - λ ) 2 + ω 2 = 0 = λ 1 , 2 = α ± . For matrix D , we have Δ( λ ) = λ 2 + ω 2 n = 0 = λ 1 , 2 = ± n . (b) Using the definition ( Av = λ v ), we can find three linearly independent eigenvectors corresponding to these distinct eigenvalues. One such set is: v 1 = 1 0 . 75 0 . 5 ; v 2 = 1 0 . 5 0 . 5 ; v 3 = 1 0 . 6667 0 . 3337 Given these linearly independent eigenvectors, we can form a nonsingular transformation that will diagonalize the matrix A . Since matrix B has repeated eigenvalues, we need to investigate whether or not we can find three linearly independent eigenvectors associated with these repeated eigenvalues. To do this check that ρ ( B - 2 I ) = 1 = γ ( B - 2 I ) = 2 . Since the nullity of the matrix ( B - 2 I ) is two, we conclude that we can only find two linearly independent eigenvectors for this matrix. Therefore, this matrix is not diagonalizable. C is diagonalizable since it has distinct eigenvalues. D is diagonilizable since it has distinct eigenvalues. 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.
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ENSC 483 HW#3 Solution–M. Saif 2 (d) What do you conclude from this exercise? Solution:
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e483h3s_07 - 1 ENSC 483 HW#3 Solution–M Saif Simon Fraser...

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