College Algebra Exam Review 399

College Algebra Exam Review 399 - 409 8.7 JORDAN CANONICAL...

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Unformatted text preview: 409 8.7. JORDAN CANONICAL FORM The companion matrix of p.x/ is the 1–by–1 matrix Œ , and N is the 1–by–1 matrix Œ1. The matrix of T1 with respect to B is 2 3 00 00 61 0 0 07 6 7 0 07 60 1 6: : : 7 Jm . / D 6 : : :: ::: : : 7 : : :7 :: 6: : 6 7 40 0 0 : : : 05 000 1 Definition 8.7.1. The matrix Jm . / is called the Jordan block of size m with eigenvalue . Lemma 8.7.2. Let T be a linear transformation of a vector space V over a field K . V has an ordered basis with respect to which the matrix of T is the Jordan block Jm . / if, and only if, V is a cyclic KŒx–module with period .x /m . Proof. We have seen that if V is a cyclic KŒx–module with generator v0 and period .x /m , then B D .v0 ; .T /v0 ; : : : ; .T /m 1 v0 / is an ordered basis of V such that ŒT B D Jm . /. Conversely, suppose that B D .v0 ; v1 ; : : : ; vm / is an ordered basis of V such that ŒT B D Jm . /. The matrix of T with respect to B is the Jordan block Jm .0/ with zeros on the main diagonal. It follows that .T /k v0 D vk for 0 Ä k Ä m 1, while .T /m v0 D 0. Therefore, V is cyclic with generator v0 and period .x /m . I Suppose that the characteristic polynomial of T factors into linear factors in KŒx. Then the elementary divisors of T are all of the form .x /m . Therefore, in the elementary divisor decomposition .T; V / D .T1 ; V1 / ˚ ˚ .T t ; V t /; each summand Vi has a basis with respect to which the matrix of Ti is a Jordan block. Hence V has a basis with respect to which the matrix of T is block diagonal with Jordan blocks on the diagonal. ...
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