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College Algebra Exam Review 400

# College Algebra Exam Review 400 - 410 8 MODULES Deﬁnition...

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Unformatted text preview: 410 8. MODULES Deﬁnition 8.7.3. A matrix is said to be in Jordan canonical form if it is block diagonal with Jordan blocks on the diagonal. 2 3 Jm1 . 1 / 0 0 60 Jm2 . 2 / 07 6 7 6 7: : : :: : : 4 : : : 05 0 0 Jm t . t / Theorem 8.7.4. (Jordan canonical form for a linear transformation.) Let T be a linear transformation of a ﬁnite dimensional vector space V over a ﬁeld K . Suppose that the characteristic polynomial of T factors into linear factors in KŒx. (a) V has a basis with respect to which the matrix of T is in Jordan canonical form. (b) The matrix of T in Jordan canonical form is unique, up to permutation of the Jordan blocks. Proof. We have already shown existence of a basis B such that ŒT B is in Jordan canonical form. The proof of uniqueness amounts to showing that a matrix in Jordan canonical form determines the elementary divisors of T . In fact, suppose that the matrix A of T with respect to some basis is in Jordan canonical form, with blocks Jmi . i / for 1 Ä i Ä t . It follows that .T; V / has a direct sum decomposition .T; V / D .T1 ; V1 / ˚ ˚ .T t ; V t /; where the matrix of Ti with respect to some basis of Vi is Jmi . i /. By mi Lemma 8.7.2, Vi is a cyclic KŒx–module with period .x i / . By uniqueness of the elementary divisor decomposition of V , Theorem 8.5.16, the polynomials .x i /mi are the elementary divisors of the KŒx–module V , that is, the elementary divisors of T . Thus, the blocks of A are uniquely determined by T , up to permutation. I Deﬁnition 8.7.5. A matrix is called the Jordan canonical form of T if it is in Jordan canonical form, and it is the matrix of T with respect to some basis of V . The Jordan canonical form of T is determined only up to permutation of its Jordan blocks. ...
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