College Algebra Exam Review 400

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

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: 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. ...
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