s11_mthsc853_hw08 - MthSc 853(Linear Algebra Dr Macauley HW...

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MthSc 853 (Linear Algebra) Dr. Macauley HW 8 Due Wednesday, March 16th, 2011 (1) Consider the following matrices: A = - 2 0 0 - 2 , B = 0 - 2 - 2 0 , C = 3 1 - 1 1 . (a) Determine the characteristic and minimal polynomials of A , B , and C . (b) Determine the eigenvectors and generalized eigenvectors of A , B , and C . (2) Consider the following matrices: A = 2 - 2 14 0 3 - 7 0 0 2 , B = 0 - 4 85 1 4 - 30 0 0 3 , C = 2 2 1 0 2 - 1 0 0 3 . A straightforward calculation shows that the characteristic polynomials are p A ( t ) = p B ( t ) = p C ( t ) = ( t - 2) 2 ( t - 3) . (a) Determine the minimal polynomials m A ( t ), m B ( t ), and m C ( t ). (b) Determine the eigenvectors and generalized eigenvectors of A , B , and C . (c) Determine which of these matrices are similar. (3) Compute the Jordan canonical form of the following matrices: A = - 1 0 1 0 2 1 2 1 0 0 - 1 0 4 0 - 6 1 , B = 1 0 0 1 2 1 0 - 4 1 0 1 - 2 0 0 0 1 . (4) Let λ be an eigenvalue of A , and let N i be the nullspace of ( A - λI ) i . Prove that A - λI extends to a well-defined map N i +1 /N i -→ N i /N i - 1 , and that this mapping is 1–1.
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