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Unformatted text preview: 398 8. MODULES every matrix in the similarity class. If two matrices have the same rational canonical form A , then they are both similar to A and therefore similar to each other. n Corollary 8.6.9. Suppose K F are two fields and A;B are two matrices in Mat n .K/ . (a) The rational canonical form of A in Mat n .F/ is the same as the rational canonical form of A in Mat n .K/ . (b) A and B are similar in Mat n .F/ if, and only if, they are similar in Mat n .K/ . Proof. The similarity class (or orbit) of A in Mat n .K/ is contained in the similarity orbit of A in Mat n .F/ , and each orbit contains exactly one matrix in rational canonical form. Therefore, the rational canonical form of A in Mat n .F/ must coincide with the rational canonical form in Mat n .K/ . If A and B are similar in Mat n .K/ , they are clearly similar in Mat n .F/ . Conversely, if they are similar in Mat n .F/ , then they have the same ratio nal canonical form in Mat n .F/ . By part (a), they have the same rational canonical form in Mat...
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 Fall '08
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 Algebra, Matrices

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