Re: repeated eigenvalues

From: Dave Rusin (rusin_at_vesuvius.math.niu.edu)
Date: 01/30/05 

Date: 30 Jan 2005 19:51:27 GMT

In article <1107112299.285024.16660@c13g2000cwb.googlegroups.com>,
 <carlos@colorado.edu> wrote:
>Jeremy Watts wrote:
>> is there any way of telling whether a matrix will have repeated eigenvalues?
>> and if so, which of the eigenvalues found are the repeated ones?
>
>For a general matrix there are no a priori tests I am aware of.

In principle, if the entries of the matrix are known exactly, one
need "only" compute gcd( det(xI - M), (d/dx) det(xI-M) ).
Of course while derivative and gcd computations are fast, the
computation of the characteristic polynomial is slow. But that prompts
the question: is there a way to compute the gcd (or otherwise find the
multiple eigenvalues) efficiently using matrix techniques which are more
efficient than computing the characteristic polynomial?
(Assume rational or symbolic entries to eliminate concerns with round-off
of floating-point calculations.)

dave