College Algebra Exam Review 403

College Algebra Exam Review 403 - , J is similar to J t ....

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
8.7. JORDAN CANONICAL FORM 413 Lemma 8.7.11. J m .±/ is similar to its transpose. Proof. Write J t m .±/ for the transpose of J m .±/ . We have J m .±/ D ±E C J m .0/ , and J t m .±/ D ±E C J t m .0/ . Therefore PJ m .±/P D J t m .±/ . n Lemma 8.7.12. Let A D A 1 ˚±±±˚ A s and B D A 1 ˚±±±˚ B s be block diagonal matrices. (a) A t D A t 1 ˚ ±±± ˚ A t s . (b) If A i is similar to B i for each i , then A is similar to B . Proof. Exercise. n Lemma 8.7.13. A matrix in Jordan canonical form is similar to its trans- pose. Proof. Follows from Lemmas 8.7.11 and 8.7.12 . n Proof of Proposition 8.7.9 : Let A 2 Mat n .K/ . Note that the characteristic polynomial of A is the same as the characteristic polynomial of A t . Let F be a field containing K such that ² A .x/ factors into linear factors in FŒxŁ . By xxx, it suffices to show that A and A t are similar in Mat n .F/ . In the following, similarity means similarity in Mat n .F/ . A is similar to its Jordan form J , and this implies that A t is similar to J t . By Lemma 8.7.13
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: , J is similar to J t . So, by transitivity of similarity, A is similar to A t . Computing the Jordan canonical form We will present two methods for computing the Jordan canonical form of a matrix. First method. The rst method is the easier one for small matrices, for which computations can be done by hand. The method is based on the following observation. Suppose that T is linear operator on a vector space V and that V is a cyclic Kx module with period a power of .x / . Then V has (up to scalar multiples) a unique eigenvector x with eigenvalue . We can successively solve for vectors x 1 ;x 2 ;::: satisfying .T /x 1 D x , .T /x 2 D x 1 , etc. We nally come to a vector x r such...
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