Unformatted text preview: n ŁP ± 1 D Œy 1 ;:::;y n ± s ;z 1 ;:::;z s Ł: This is a basis of F over KŒxŁ , and Œy 1 ;:::;y n ± s ;z 1 ;:::;z s ŁD.x/ D Œy 1 ;:::;y n ± s ;a 1 .x/z 1 ;:::;a s .x/z s Ł is a basis of ker .˚/ . It follows that f v 1 ;:::;v s g WD f ˚.z 1 /;:::;˚.z s / g are the generators of cyclic subspaces V 1 ;:::;V s of V , such that V D V 1 ˚³³³˚ V s , and v j has period a j .x/ . One calculates these vectors with the aid of T : if P ± 1 D .b i;j .x// , then z j D X i b i;n ± s C j .x/f i ; so v j D X i b i;n ± s C j .T /e i : Let ı j denote the degree of a j .x/ . Then .v 1 ;T v 1 ;:::;T ı 1 ± 1 v 1 I v 2 ;T v 2 ;:::;T ı 2 ± 1 v 2 I :::/ is a basis of V with respect to which the matrix of T is in rational canonical form. The reader is asked to ﬁll in some of the details of this discussion in Exercise 8.6.3 ....
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 Fall '08
 EVERAGE
 Linear Algebra, Algebra, Polynomials, Addition, Invertible matrix, Elementary matrix, KŒx, compute invertible matrices

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