solution9 - which is represented by the matrix A in the...

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Homework assignment 9 Section 6.2 pp. 189 Exercise 1. In each of the following cases, let T be the linear operator on R 2 which is represented by the matrix A in the standard ordered basis for R 2 , and let U be the linear operator on C 2 represented by A in the standard ordered basis. Find the characteristic polynomial for T and that for U , find the characteristic values of each operator, and for each such characteristic value c find a basis for the corresponding space of characteristic vectors. A = ± 1 0 0 0 A = ± 2 3 - 1 1 A = ± 1 1 1 1 Solution: In all cases, denote by B c the basis for the subspace corresponding to the characteristic value c First matrix The characteristic polynomial is x ( x - 1) . The roots are 0 and 1 . B 0 = { (0 , 1) } and B 1 = { (1 , 0) } . In this case the real and complex cases are the same. Second matrix The characteristic polynomial is 5 - 3 x + x 2 which has no real roots. The complex eigenvalues are 3 2 + i 1 2 11 , 3 2 - i 1 2 11 , B 3 2 + i 1 2 11 = { (1 , - 1 6 + i 1 6 11) } and B 3 2 - i 1 2 11 = { (1 , - 1 6 - i 1 6 11) } Third matrix The characteristic polynomial is
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This note was uploaded on 11/14/2010 for the course PHYCS 498 taught by Professor Aa during the Spring '10 term at University of Illinois, Urbana Champaign.

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solution9 - which is represented by the matrix A in the...

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