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At 11:44 AM 11/8/2006, Chuck, you wrote:
So if the eigenvectors can be complex even when one component is
chosen to be real, then that implies that the oscillators in a normal
mode need NOT pass through the equilibrium position at the same
instant (ie. relative phases other than 0 or pi are possible).
In that case, can someone rig up a *simple soluble* example of such
an eigenvalue problem?
To make my request clear for those who have forgotten the beginning
of this thread. I want an example of a system of coupled oscillators,
as simple as possible, where the relative phases between oscillators
in a normal mode are neither 0 nor pi. All the usual textbook
examples I can think only have in-phase or exactly-out-of-phase
relative motions of the oscillators.
[Jack]
The correct statement, that you seem to be striving for, is that the......
eigenvalues of a Hermitean matrix (includes real, symmetric
matrices) are real. This says nothing about the scaling of the
eigenvectors, which are usually scaled by normalizing them to a
(squared) norm of unity - which leaves them with an undetermined
phase. That phase is just the subject of the present discussion.
Regards,
Jack
Is that true even if I insist that one component arbitrarily have the
real value 1, to remove the indeterminacy in the overall scaling of
an eigenvector? This way d_i is the relative phase, which is what I
meant.
The argument is erroneous. The reality condition is on the
eigenvalues, not the eigenvectors.
Jack
--
Carl E Mungan, Assoc Prof of Physics 410-293-6680 (O) -3729 (F)
It's all very well for Jack to mention the eigenvalue of the NxN
Jacobi matrices
- equivalent to estimating the largest zero of a sequence of
orthogonal polynomials.
One could, I suppose, go further in systematizing properties of matrix model
partition functions which I define here as solutions to the
Virasoro-like sets of
linear differential equations and proceed to consideration of non-Gaussian
phases of the Hermitean one-matrix model.
The underlying concept, dimly limned, is about connected correlators
like phase-independent "check-operators" acting on the small space
of T-variables which parameterize the polynomial W(z). These multidensity
check-operators with the appropriate reservations, look very similar
to the Gaussian case. This suggestion is speculative however, and may
even be a figment of my imagination.
Here is a URL, not for a physical system of chaotic phase to which
Jack refers and for which you seek examples, but rather the usual
pi phase coupling to which I previously alluded.
<http://facstaff.morehouse.edu/~cmoore/CoupledOscillations.htm>
Brian Whatcott Altus OK Eureka!
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