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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 is a root of T .x/ if, and only if, T has an eigenvector in V with eigenvalue . Show that v is an eigenvector of T for some eigen-value if, and only if, the one dimensional subspace Kv V is invariant under T . The next four exercises give an alternative proof of the Cayley-Hamilton theorem. Let T 2 End K .V / , where V is n dimensional. Assume that the eld K contains all roots of T .x/ ; that is, T .x/ factors into linear factors in Kx . 8.6.8. Let V V be any invariant subspace for T . Show that there is a linear operator T on V=V dened 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