College Algebra Exam Review 409

College Algebra Exam Review 409 - A , and nd a basis of K n...

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8.7. JORDAN CANONICAL FORM 419 (c) Show that the trace of N is zero. (d) Show that the Jordan canonical form of N is the same as the rational canonical form of N . 8.7.7. Classify nilpotent matrices in Mat n .K/ up to similarity. Hint: What are the possible Jordan canonical forms? 8.7.8. Let K be a field of arbitrary characteristic, and suppose that ± is a primitive n -th root of unity in K ; that is ± n D 1 , and ± s ¤ 1 for any s < n . Let S denote the n –by– n permutation matrix corresponding to the permutation .1;2;3; ±±± ;n/ . For example, for n D 5 , S D 2 6 6 6 6 4 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 3 7 7 7 7 5 (a) Show that S is similar in Mat n .K/ to the diagonal matrix D with diagonal entries 1;±;± 2 ;:::;± n ± 1 . (b) Conclude that S and D have the same trace, and therefore 1 C ± C ± 2 C ±±± C ± n ± 1 D 0: 8.7.9. Let A denote the 10 –by– 10 matrix over a field K with all entries equal to 1 . (a) If the characteristic of K is not 2 or 5 , show that A is diagonaliz- able, find the (diagonal) Jordan canonical form of
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Unformatted text preview: A , and nd a basis of K n consisting of eigenvectors of A . (b) If the characteristic of K is 2 or 5 , show that A is nilpotent, nd the Jordan canonical form of A , and nd an invertible S such that S 1 AS is in Jordan canonical form. 8.7.10. Let A denote the 10 by 10 matrix over a eld K with all entries equal to 1 , except the diagonal entries, which are equal to 4. (a) If the characteristic of K is not 2 or 5 , show that A is diagonaliz-able, nd the (diagonal) Jordan canonical form of A , and nd a basis of K n consisting of eigenvectors of A . (b) If the characteristic of K is 2 or 5 , nd the Jordan canonical form of A , and nd an invertible S such that S 1 AS is in Jordan canonical form....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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