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College Algebra Exam Review 397

College Algebra Exam Review 397 - 8.7 JORDAN CANONICAL FORM...

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Unformatted text preview: 8.7. JORDAN CANONICAL FORM 407 8.7. Jordan Canonical Form We continue with the analysis of the previous section. If T is a linear transformation of a finite dimensional vector space V over a field K , then V has the structure of a finitely generated torsion module over KOExŁ , de- fined by f.x/v D f.T/v for f.x/ 2 KOExŁ . A decomposition of V as a direct sum of KOExŁ –submodules is the same as decomposition as a direct sum of T –invariant linear subspaces. The rational canonical form of T corresponds to the invariant factor decomposition of the KOExŁ –module V . We now consider the elementary divisor decomposition of V (which may be regarded as a refinement of the invariant factor decomposition). The elementary divisor decomposition (Theorem 8.5.16 ) displays V as the direct sum of cyclic submodules, each with period a power of some monic irreducible polynomial in KOExŁ . That is, each direct summand is isomor- phic to KOExŁ=.p.x/ m / for some monic irreducible p.x/ and some...
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