# S.O.S. Mathematics CyberBoard

Your Resource for mathematics help on the web!
 It is currently Wed, 22 May 2013 03:20:34 UTC

 All times are UTC [ DST ]

 Page 1 of 1 [ 5 posts ]
 Print view Previous topic | Next topic
Author Message
 Post subject: Linear mapPosted: Mon, 16 Feb 2009 14:07:32 UTC
 S.O.S. Oldtimer

Joined: Thu, 30 Oct 2008 10:47:09 UTC
Posts: 182
Let be a complex vector space and let be a linear map satisfying .
Prove that has a basis consisting of eigenvectors of the map .

Any idea ?

Top

 Post subject: Re: Linear mapPosted: Mon, 16 Feb 2009 14:27:50 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Beta wrote:
Let be a complex vector space and let be a linear map satisfying .
Prove that has a basis consisting of eigenvectors of the map .

Any idea ?

Hint: , where .

Top

 Post subject: Posted: Mon, 16 Feb 2009 15:13:53 UTC
 S.O.S. Oldtimer

Joined: Thu, 30 Oct 2008 10:47:09 UTC
Posts: 182
So the eigenvalues are , for some , with multiplicity .
And the eigenvectors can be any non-zero vectors?

Top

 Post subject: Posted: Mon, 16 Feb 2009 15:50:03 UTC
 Moderator

Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 6007
Location: 127.0.0.1, ::1 (avatar courtesy of UDN)
Beta wrote:
So the eigenvalues are , for some , with multiplicity .
And the eigenvectors can be any non-zero vectors?

Err... you can't deduce the algebraic&geometric multiplicities from that. However, depending on what you know, either you play around with that equation and show that , or you invoke the minimal polynomial is a product of distinct linears iff diagonalisable. Both will give you what you want.

Top

 Post subject: Re: Linear mapPosted: Mon, 16 Feb 2009 22:26:53 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 12098
Location: Austin, TX
Beta wrote:
Let be a complex vector space and let be a linear map satisfying .
Prove that has a basis consisting of eigenvectors of the map .

Any idea ?

Well, we have that , think characteristic polynomial, and since it splits, what does that say about diagonalizability?

_________________
(\ /)
(O.o)
(> <)
This is Bunny. Copy Bunny into your signature to help him on his way to world domination

Top

 Display posts from previous: All posts1 day7 days2 weeks1 month3 months6 months1 year Sort by AuthorPost timeSubject AscendingDescending
 Page 1 of 1 [ 5 posts ]

 All times are UTC [ DST ]

#### Who is online

Users browsing this forum: No registered users

 You cannot post new topics in this forumYou cannot reply to topics in this forumYou cannot edit your posts in this forumYou cannot delete your posts in this forum

Search for:
 Jump to:  Select a forum ------------------ High School and College Mathematics    Algebra    Geometry and Trigonometry    Calculus    Matrix Algebra    Differential Equations    Probability and Statistics    Proposed Problems Applications    Physics, Chemistry, Engineering, etc.    Computer Science    Math for Business and Economics Advanced Mathematics    Foundations    Algebra and Number Theory    Analysis and Topology    Applied Mathematics    Other Topics in Advanced Mathematics Other Topics    Administrator Announcements    Comments and Suggestions for S.O.S. Math    Posting Math Formulas with LaTeX    Miscellaneous