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 20 . (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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