College Algebra Exam Review 402

College Algebra Exam Review 402 - Any matrix in Mat n .K/...

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412 8. MODULES Proposition 8.7.8. Let A and B be two matrices in Mat n .K/ with the same characteristic polynomial ±.x/ . Suppose that F ± K is an extension field such that ±.x/ factors into linear factors in FŒxŁ . Then A and B are similar in Mat n .K/ if, and only if, they have the same Jordan canonical form in Mat n .F/ (up to permutation of Jordan blocks). Proof. A and B are similar in Mat n .K/ if, and only if, they are similar in Mat n .F/ by Corollary 8.6.9 , and they are similar in Mat n .F/ if, and only if, they have the same rational canonical form by Proposition 8.6.8 . But each of the following invariants of A in Mat n .F/ determines all the others: the rational canonical form, the invariant factors, the elementary divisors, and the Jordan canonical form. n Let us give a typical application of these ideas to matrix theory. Proposition 8.7.9.
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Unformatted text preview: Any matrix in Mat n .K/ is similar to its transpose. The idea is to show that the assertion holds for a Jordan block and then to use the theory of canonical forms to show that this special case implies the general case. Lemma 8.7.10. J m .0/ is similar to its transpose. Proof. Write J t m .0/ for the transpose of J m .0/ . J m .0/ acts as follows on the standard basis vectors: J m .0/ W O e 1 7! O e 2 7! ²²² 7! O e m ± 1 7! O e m 7! 0; while J t m .0/ acts as follows: J t m .0/ W O e m 7! O e m ± 1 7! ²²² 7! O e 2 7! O e 1 7! 0; If P is the permutation matrix that interchanges the standard basis vector as follows: O e 1 $ O e m ; O e 2 $ O e m ± 1 ; and so forth ; then we have P 2 D E and PJ m .0/P D J t m .0/: n...
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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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