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Left eigenvectors?

  • To: mathgroup at
  • Subject: [mg1008] Left eigenvectors?
  • From: john.r.manuel at Dartmouth.EDU (John Manuel)
  • Date: Mon, 8 May 1995 05:48:29 -0400
  • Organization: Dartmouth College, Thayer School of Engineering


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),

  rvec = {r1, r2, r3, r4, r5, r6, r7, r8};
  Solve[A1.rvec == val*rvec, rvec]
  {{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

Help and pointers would be much appreciated.


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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