College Algebra Exam Review 387

College Algebra Exam Review 387 - 8.6. RATIONAL CANONICAL...

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Unformatted text preview: 8.6. RATIONAL CANONICAL FORM 397 and only if, the KŒx–modules determined by these linear transformations are isomorphic as KŒx–modules. Let V1 denote V endowed with the KŒx–module structure derived from T1 and let V2 denote V endowed with the KŒx–module structure derived from T2 . Suppose U W V1 ! V2 is a KŒx–module isomorphism; then U is a vector space isomorphism satisfying T2 .U v/ D x.U v/ D U.xv/ D U.T1 v/. It follows that T2 D U T1 U 1 . Conversely, suppose that U is an invertible linear transformation such that T2 D U T1 U 1 . It follows that for all f .x/ 2 KŒx, f .T2 / D Uf .T1 /U 1 ; equivalently, f .T2 /U v D Uf .T1 /v for all v 2 V But this means that U is a KŒx–module isomorphism from V1 to V2 . I Rational canonical form for matrices Let A be an n–by–n matrix over a field K . Let T be the linear transformation of K n determined by left multiplication by A, T .v/ D Av for v 2 K n . Thus, A is the matrix of T with respect to the standard basis of K n . A second matrix A0 is similar to A if, and only if, A0 is the matrix of T with respect to some other ordered basis. Exactly one such matrix is in rational canonical form, according to Theorem 8.6.4. So we have the following result: Proposition 8.6.6. Any n–by–n matrix is similar to a unique matrix in rational canonical form. Definition 8.6.7. The unique matrix in rational canonical form that is similar to a given matrix A is called the rational canonical form of A. The blocks of the rational canonical form of A are companion matrices of monic polynomials a1 .x/; : : : ; as .x/ such that ai .x/ divides aj .x/ if i Ä j . These are called the invariant factors of A. The rational canonical form is a complete invariant for similarity of matrices. Proposition 8.6.8. Two n–by–n matrices are similar in Matn .K/ if, and only if, they have the same rational canonical form. Proof. There is exactly one matrix in rational canonical form in each similarity equivalence class, and that matrix is the rational canonical form of ...
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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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