College Algebra Exam Review 397

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 KOEx , de- fined by f.x/v D f.T/v for f.x/ 2 KOEx . A decomposition of V as a direct sum of KOEx 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 KOEx 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 KOEx . That is, each direct summand is isomor- phic to KOEx=.p.x/ m / for some monic irreducible p.x/ and some...
View Full Document

This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

Ask a homework question - tutors are online