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Re: Re: Numerical Left Eigenvectors


At 01:21 15-03-97 -0500, seanross at worldnet.att.net wrote:
>Gregory Dwyer wrote:
>> 
>> Greetings -
>> 
>> When I use the "eigenvector" function to numerically calculate
>> eigenvectors, Mathematica seems to assume that I always want right
>> eigenvectors.  Is there some way to numerically calculate left eigenvectors?
>> 
>> Thanks.
>> 
>> Greg Dwyer.
>> Entomology, UMASS Amherst
>> dwyer at ent.umass.edu
>
>
>What is a left eigenvector?  The only eigenvectors I know of are ones 
>that satisfy the following equation:
>
>Matrix . vector = constant vector.  Is this a right eigenvector?  Does 
>left eigenvector mean
>vector. Matrix = constant vector?
>
>If so, have you looked for a way to express Matrix . column vector in 
>terms of row vector . Matrix?  For a 2x2 case, it looks like 
>Matrix.columnvector = rowvector.TransposeMatrix
>
>I am not sure how commutable square matrices and vectors are supposed to 
>be.  Perhaps someone else out there with experience in something like 
>group theory or who has a copy of MathTensor etc. could enlighten you 
>further.
>
>
>

hi,

straight from The (electronic) Book under 'Eigensystem':

In[102]:=
it
Out[102]=
{{2,3,4,5},{1,2,7,8},{6,9,2,1}}
In[104]:=
mit=Join[it,{{3,3,9,9}}]
Out[104]=
{{2,3,4,5},{1,2,7,8},{6,9,2,1},{3,3,9,9}}
In[135]:=
{vals,vecs}=Eigensystem[N at mit];
vals
vecs//MatrixForm
Out[135]=
{19.1216,-2.27556+1.89411 I,-2.27556-1.89411 I,0.429553}
Out[136]//MatrixForm=
 *** stuff clipped ***
In[137]:=
mit.Transpose[vecs] == Transpose[vecs].DiagonalMatrix[vals]
Out[137]=
True
In[138]:=
(vecs.Transpose[mit] //Chop)==(DiagonalMatrix[vals].vecs//Chop)
Out[138]=
True

beware : without Chop no equality! different order of operations gives tiny
rounding errors. Except for pure Integers of course.

wouter.

Dr. Wouter L. J. MEEUSSEN
eu000949 at pophost.eunet.be
w.meeussen.vdmcc at vandemoortele.be



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