131F06.probsess1

131F06.probsess1 - Compute A 2 . Without doing any further...

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Problem Session for MTH G131: 10/12/06 1). Find eigenvalues and eigenvectors of the matrix A = ± 3 - 5 2 - 3 ² Use your result to find the solution of the following ODE: d dt ± x y ² = A ± x y ² , ± x (0) y (0) ² = ± 1 2 ² 2). For a 2 × 2 matrix A with eigenvalues λ 1 and λ 2 show that Trace( A ) = λ 1 + λ 2 , Det( A ) = λ 1 λ 2 3). Suppose the 2 × 2 matrix A has distinct real eigenvalues 0 < λ 1 < λ 2 , with eigenvectors v 1 and v 2 . Show that for most vectors w lim n →∞ ³ λ - 1 2 A ´ n w c v 2 for some constant c . For which vectors w is this result false? 4). Consider the 2 × 2 matrix A = ± 0 1 1 0 ²
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Unformatted text preview: Compute A 2 . Without doing any further calculations, what can you con-clude about the eigenvalues of A ? 5). Consider the 3 3 matrix B = 1 1 1 Compute B 3 . Without doing any further calculations, what can you con-clude about the eigenvalues of B ? 6). Suppose the matrix A satises the equation A 4-3 A 3 + 6 A 2-I = 0 What can you conclude about the eigenvalues of A ?...
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This note was uploaded on 10/19/2011 for the course MATH 3354 taught by Professor Drager during the Fall '08 term at Texas Tech.

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