s11_mthsc853_hw07

s11_mthsc853_hw07 - MthSc 853 (Linear Algebra) Dr. Macauley...

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MthSc 853 (Linear Algebra) Dr. Macauley HW 7 Due Wednesday, March 9th, 2011 (1) Prove the following properties of the trace function: (a) tr AB = tr BA for all m × n matrices A and n × m matrices B . (b) tr AA T = a 2 ij for all n × n matrices A . (2) Find the eigenvalues and corresponding eigenvectors for the following matrices over C . ( a ) ± 2 4 5 3 ² ( b ) ± 3 2 - 2 3 ² ( c ) 5 - 6 - 6 - 1 4 2 3 - 6 - 4 ( d ) 3 1 - 1 2 2 - 1 2 2 0 . (3) (a) Show that if A and B are similar, then A and B have the same eigenvalues. (b) Is the converse of part (a) true? Prove or disprove. (4) Let A φ be the 3 × 3 matrix representing a rotation of R 3 through an angle φ about the y -axis. (a) Find the eigenvalues for A φ over C . (b) Determine necessary and sufficient conditions on φ in order for A φ to have 3 linearly independent eigenvectors in R 3 . Justify your claim and interpret it geometrically. (5) Let
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