Eigenvectors and eigenvalues

[Andrew Dinn]

Bonnie Surfus writes:

> I have mentioned this before and got little response.  Actually, there is 
> soemthing called an "eigenfunction" that has do with the non-differential 
> equations, attractors, and chaotica sytsems.  I am not yet sure what an 
> eigenfunction is--only saw it referenced in a book on strange attractors 
> and chaotic evolutions (by David Ruelle.)

An eigenvector is a fixed point of a linear map in a vector space -
actually the vector's direction is fixed, it's length is multiplied by
a scalar otherwise known as the `eigenvalue' assoiated with the
eigenvector. For example, the transformation which rotates 3-d vectors
45 degrees about the Z axis has vector (0 0 1) as its only eigenvector
and 1 as the corresponding eigenvalue. Another example would be the
transformation which scales up by a factor of 2 along each axis. In
this case every vector is an eigenvector with eigenvalue 2.

Presumably where the vectors are functions the eigenvector is called
an eigenfunction. Function spaces occur in quantum mechanics where the
probability functions for quantum level entities are elements of a
vector space of functions. Eigenvectors are associated with observable
variables.

P.S. In German `eigen' means `self' or `own' (it functions as an
adjective).


Andrew Dinn
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there is no map / and a compass / wouldn't help at all