Unformatted text preview: if it is block diagonal 2 6 6 6 4 C a 1 ³³³ C a 2 ³³³ : : : : : : : : : ³³³ C a s 3 7 7 7 5 ; where C a i is the companion matrix of a monic polynomial a i .x/ of degree ´ 1 , and a i .x/ divides a j .x/ for i ± j Theorem 8.6.4. (Rational canonical form) Let T be a linear transforma-tion of a ﬁnite dimensional vector space V over a ﬁeld K . (a) There is an ordered basis of V with respect to which the matrix of T is in rational canonical form. (b) Only one matrix in rational canonical form appears as the matrix of T with respect to some ordered basis of V . Proof. According to Theorem 8.5.16 , .T;V / has a direct sum decomposi-tion .T;V / D .T 1 ;V 1 / ˚ ³³³ ˚ .T s ;V s /;...
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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.
- Fall '08