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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.
 Fall '08
 EVERAGE
 Algebra, Factors, Matrices

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