Math Is Fun Forum

LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

Proof of Linear Algebra Theorem

Im reading a booklet on Methods of Applied Analysis and they mention this theorem and say the proof can be looked at in any linear algebra book. I looked in my linear algebra book and could not find such theorem.

Could anyone care to provide a link to a proof or state it here if its readily available? Thanks.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Proof of Linear Algebra Theorem

Are you familiar with Jordan canonical form?  The proof of this is to use the Jordan–Chevalley decomposition.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

juriguen

Member

Registered: 2009-07-05

Posts: 59

Re: Proof of Linear Algebra Theorem

Hi!

I have no books with me to check the demo, but I believe you can show it following a couple of links from Wikipedia:

1. Jordan normal form, http://en.wikipedia.org/wiki/Jordan_form

First of all, since A is a general matrix, it may not be diagonalizable (in this case it would happen when it doesn't have n independent eigenvectors). But then we can use its normal Jordan form:

2. Now lets work with the Jordan form J of A, and from http://en.wikipedia.org/wiki/Dunford_decomposition we know:

where D is diagonal and N is nilpotent (i.e. N^m = 0 for some integer m).

3. Then, from the above definition and the "Generalization" in http://en.wikipedia.org/wiki/Matrix_exponential:

If D commutes with N (DN = ND), we can write:

4. In this case, still following Wikipedia:

5. Now, using the previous partial results:

Anybody, please correct me.

Hope it helps, anyway

Jose

Last edited by juriguen (2009-08-04 20:39:07)

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