Unformatted text preview: the sum of the matrix entries on the main diagonal of A . Conclude that the trace is a similarity invariant. 8.6.7. Show that ² is a root of ± T .x/ if, and only if, T has an eigenvector in V with eigenvalue ² . Show that v is an eigenvector of T for some eigenvalue if, and only if, the one dimensional subspace Kv ´ V is invariant under T . The next four exercises give an alternative proof of the CayleyHamilton theorem. Let T 2 End K .V / , where V is n –dimensional. Assume that the ﬁeld K contains all roots of ± T .x/ ; that is, ± T .x/ factors into linear factors in KŒxŁ . 8.6.8. Let V ´ V be any invariant subspace for T . Show that there is a linear operator T on V=V deﬁned by T .v C V / D T.v/ C V for all v 2 V . Suppose that .v 1 ;:::;v k / is an ordered basis of V , and that .v k C 1 C V ;:::;v n C V /...
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 Fall '08
 EVERAGE
 Linear Algebra, Algebra, rational canonical form, similarity invariant, surjective KŒx–module homomorphism

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