Re: ? determine common e-vector

"William R. Frensley" <frensley@xxxxxxxxxxxx> writes:

Cheng Cosine wrote:
Hi:
Suppose both matrices A and B and N-by-N and have
their own eigenvector decomposition. Suppose some
or all of their eigenvectors are common. How do we
determine those common eigenvector(s) shared by
A and B?

Consider the concept of "simultaneously diagonalizable"
matrices, which means that two matrices can be diagonalized
by the same basis transformation. In other words, they share
the same eigenvectors. Now, simultaneously diagonalizable
matrices commute: AB - BA = 0.

If your matrices share some eigenvectors in common, I would
expect that their commutator C = [A,B] = AB-BA would reflect
this structure. The common eigenvectors should lie in the
null space of C. Therefore, I would suggest that you try
evaluating the commutator and then do SVD on the result to
identify its null space.

That's a start... but the common eigenvectors might not span all of the
null space of the commutator. You really need a subspace of that null
space that A and B map into itself.
--
Robert Israel israel@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
.

Relevant Pages

  • Re: ? determine common e-vector
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    (sci.math.num-analysis)
  • Re: ? determine common e-vector
    ... their own eigenvector decomposition. ... or all of their eigenvectors are common. ... evaluating the commutator and then do SVD on the result to ...
    (sci.math.num-analysis)
  • Re: Matrix equation
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