College Algebra Exam Review 383

College Algebra Exam Review 383 - mations T i have a simple...

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8.6. RATIONAL CANONICAL FORM 393 If V is the direct sum of several T –invariant subspaces, V D V 1 ˚ ±±± ˚ V s ; then with respect to an ordered basis that is the union of bases of the sub- spaces V i , the matrix of T has the block diagonal form: A D 2 6 6 6 4 A 1 0 ±±± 0 0 A 2 ±±± 0 : : : : : : : : : 0 0 0 ±±± A s 3 7 7 7 5 : In this situation, let T i denote the restriction of T to the invariant subspace subspace V i . In the block diagonal matrix above, A i is the matrix of T i with respect to some basis of V i . We write .T;V / D .T 1 ;V 1 / ˚ ±±± ˚ .T s ;V s /; or just T D T 1 ˚ ±±± ˚ T s to indicate that V is the direct sum of T –invariant subspaces and that T i is the restriction of T to the invariant subspace V i . We also write A D A 1 ˚ ±±± ˚ A s to indicate that the matrix A is block diagonal with blocks A 1 ;:::;A s . A strategy for understanding the structure of a linear transformation T is to find such a direct sum decomposition so that the component transfor-
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Unformatted text preview: mations T i have a simple form. Because V is a nitely generated torsion module over the Euclidean domain Kx , according to Theorem 8.5.2 , .T;V / has a direct sum decom-position .T;V / D .T 1 ;V 1 / .T s ;V s /; where V i is a cyclic Kx module V i Kx=.a i .x//; deg .a i .x// 1 (that is, a i .x/ is not zero and not a unit) and a i .x/ divides a j .x/ if i j . Moreover, if we insist that the a i .x/ are monic, then they are unique. We call the polynomials a i .x/ the invariant factors of T . To understand the structure of T , it sufces to understand how T i acts on the cyclic Kx module V i ....
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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