College Algebra Exam Review 390

College Algebra Exam Review 390 - n ŁP ± 1 D Œy 1;y n ±...

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400 8. MODULES by row and column operations. We want the diagonal entries of the result- ing matrix to be monic polynomials, but this only requires some additional row operations of type two (multiplying a row by unit in KOExŁ .) We can compute invertible matrices P and Q such that P.xE n A/Q D D.x/ D diag .1; 1; : : : ; 1; a 1 .x/; a 2 .x/; : : : ; a s .x//; where the a i .x/ are monic and a i .x/ divides a j .x/ for i j . The poly- nomials a i .x/ are the invariant factors of T , so they are all we need in order to write down the rational canonical form of T . But we can actually compute a basis of V with respect to which the matrix of T is in rational canonical form. We have xE n A D P 1 D.x/Q 1 , so OEw 1 ; : : : ; w n ŁQ 1 D OEf 1 ; : : : ; f n ŁP 1 D.x/: (Let us mention here that we compute the matrix P as a product of ele- mentary matrices implementing the row operations; we can compute the inverse of each of these matrices without additional effort, and thus we can compute P 1 without additional effort.) Set OEf 1 ; : : : ; f n ŁP
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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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