College Algebra Exam Review 391

College Algebra - rational canonical form of A is ± C a 1.x C a 2.x ² D 2 6 6 6 6 4 1 0 0 0 0 0 0 ± 8 1 0 0 20 0 1 0 ± 18 0 0 1 7 3 7 7 7 7 5

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8.6. RATIONAL CANONICAL FORM 401 Example 8.6.11. Consider the matrix A D 2 6 6 6 6 4 ± 1 0 0 0 3 1 2 0 ± 4 0 3 1 2 ± 4 ± 3 0 0 0 1 0 ± 2 0 0 0 4 3 7 7 7 7 5 2 Q ŒxŁ We compute the rational cononical form of A and an invertible matrix S such that S ± 1 AS is in rational canonical form. (Let T denote the linear transformation of Q 5 determined by multiplication by A , The columns of S form a basis of Q 5 with respect to which the matrix of T is in rational canonical form.) Using the algorithm described in the proof of Proposition 8.4.6 , we compute the Smith normal form of xE 6 ± A in Q ŒxŁ . That is we compute invertible matrices P;Q 2 Mat 6 . Q ŒxŁ/ 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 ² j . 2 The result is D.x/ D 2 6 6 6 6 4 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 ± 1 C x 0 0 0 0 0 . ± 2 C x/ 3 . ± 1 C x/ 3 7 7 7 7 5 : Therefore, the invariant factors of A are a 1 .x/ D x ± 1 and a 2 .x/ D . ± 2 C x/ 3 . ± 1 C x/ D x 4 ± 7x 3 C 18x 2 ± 20x C 8 . Consequently, the
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Unformatted text preview: rational canonical form of A is ± C a 1 .x/ C a 2 .x/ ² D 2 6 6 6 6 4 1 0 0 0 0 0 0 ± 8 1 0 0 20 0 1 0 ± 18 0 0 1 7 3 7 7 7 7 5 : Now we consider how to find a basis with respect to which the transfor-mation T determined by multiplication by A is in rational canonical form. Q 5 D V 1 ˚ V 2 , where each of V 1 and V 2 is invariant under T and cyclic for T . The subspace V 1 is one–dimensional and the subspace V 2 is four– dimensional. We obtain cyclic vectors for these two subspaces using the 2 Examples of computations of rational canonical form can be found in the notebook Canonical-Form-Examples.nb , also available on my webpage....
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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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