Re: Re: Numerical Left Eigenvectors

*To*: mathgroup at smc.vnet.net*Subject*: [mg6406] Re: [mg6392] Re: [mg6369] Numerical Left Eigenvectors*From*: "w.meeussen" <w.meeussen.vdmcc at vandemoortele.be>*Date*: Sun, 16 Mar 1997 19:25:22 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

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