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MathGroup Archive 1995

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Re: Left eigenvectors?

  • To: mathgroup at christensen.cybernetics.net
  • Subject: [mg1017] Re: Left eigenvectors?
  • From: danl (Daniel Lichtblau)
  • Date: Wed, 10 May 1995 03:35:20 -0400
  • Organization: Wolfram Research, Inc.

  For left eigenstuff, you can use Eigensystem[Transpose[A]].

  The difficulty you have with Solve stems from the fact that the system  
is overdetermined. It appears that you gave it an eigenvalue of  
multiplicity three, and it returned an eigenvector parametrized by its  
last three components.

  Daniel Lichtblau, WRI


In article <3o9mr0$b92 at news0.cybernetics.net> john.r.manuel at Dartmouth.EDU  
(John Manuel) writes:
> Hi,
> 
> I've been using Eigensystem[A] for a while now to find the eigenvalues  
and
> eigenvectors of the matrix A. The eigenvectors that it finds are the  
right
> eigenvectors of A, i.e.,
> 
>   A.rvec == val*rvec,
> 
> where val is an eigenvalue and rvec is the associated right eigenvector.
> I'd now like to find the left eigenvectors of A, i.e.,
> 
>   lvec.A == val*lvec.
> 
> How should I go about this?
> 
> I've been experimenting with Solve[], but the results are rather untidy,
> e.g. (testing the method using the right eigenvector again),
> 
> In[]:=
>   rvec = {r1, r2, r3, r4, r5, r6, r7, r8};
>   Solve[A1.rvec == val*rvec, rvec]
> Out[]:=
>   {{r3 -> 0, r4 -> 0, r5 -> -(bX*r6) - bY*r7 - bZ*r8, r2 -> 0}}
> 
> Is there a way I can indicate to Solve[] that rvec is an eight element
> vector and that it should find the value (or values) of the entire  
vector?
> Specifying rvec as I have above finds relationships between the elements
> of the vector but what I want are the eigenvectors, such as those found  
by
> Eigensystem[].
> 
> Help and pointers would be much appreciated.
> 
> John
> 
> ----
> John R. Manuel        office: (603) 646-2723        fax: (603) 646-3856
> Dartmouth College, Thayer School of Engineering, Hanover, NH, 03755 USA
> 
> 


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