wksht12 - MATH 152 L1 & L2 Applied Linear Algebra and...

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Unformatted text preview: MATH 152 L1 & L2 Applied Linear Algebra and Differential Equations Week 12 Worksheet (April 25, 2005): Q1. (Eigenvalues and eigenvectors) Spring 2004-05 Eigenvalue and Eigenvector, Differential Systems (a) Find an example of 3 × 3 matrices A and B, such that A and B have the same eigenvectors but distinct eigenvalues. (b) Find an example of 3 × 3 matrices A and B, such that A and B have the same eigenvalues but distinct eigenvectors. Q2. (Diagonalizable matrix) Consider 10000 2 2 0 0 0 A = 3 3 3 0 0 , 4 4 4 4 0 55555 1 0 P = 0 0 0 0 .1 1 0 0 0 0.01 10 e 0 0 0.001 100 e2 π 0 0.0001 1000 e3 √ . π 1 Is A diagonalizable? Find the eigenvalues of P−1 A2005 P. 0 Q3. (Diagonalizable matrix) Find eigenvalues and eigenvectors of the matrix A = 0 1 Write down the diagonalization of A if it is diagonalizable. 0 1 0 −1 0 . 0 2 Q4. (Diagonalizable matrix) Find eigenvalues and eigenvectors of the matrix A = −5 −2 Write down the diagonalization of A if it is diagonalizable. 2 1 4 −2 2 . −1 1 Q5. (Non-diagonalizable matrix) Find eigenvalues and eigenvectors of the matrix A = 0 0 Write down the diagonalization of A if it is diagonalizable. 2 1 0 3 2 . 1 Q6. (Differential system – distinct real eigenvalues) Solve the differential system x (t) = 5 4 −2 x(t). −1 Q7. (Differential system – complex eigenvalues) 1 x (t) = 0 2 Solve the differential system 1 −2 1 −4 x(t). 0 −1 Q8. (Nonhomogeneous system) Solve the differential system x (t) = 1 2 3 t+1 x(t) + . 2 t2 ...
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This note was uploaded on 09/30/2010 for the course MATH MATH152 taught by Professor Kcc during the Spring '10 term at HKUST.

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wksht12 - MATH 152 L1 & L2 Applied Linear Algebra and...

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