On Fri, 18 Feb 2000, Falk H. Koenemann wrote:
> Wojciech Czaplinski schrieb:
> > Hello All,
> > Can anyone explain to me (or give some references explaining) what the
> > hell are eigenvalues and eigenvectors ??? Bear in mind, that my maths
> > neurons got severely damaged by a heavy case of boreliosis ;)
> >
> > thanks in advance - Wojtek
>
I tried reading the answer by Falk, but to be honest I didn't
understand a thing. And I'm a mathematician :-)
Eigenvalues and eigenvectors are mathematical concepts.
While they may have applications elsewhere, let's not
mix up definitions (the `real' meaning) and interpretations.
Take a square matrix, A (say with n rows and n columns).
Then an n-vector v (i.e. a vector with n components) and
a number lambda are an eigenvector/eigenvalue pair for A, if
A . v = lambda . v
The first dot is a matrix-vector multiplication, the second
just a multiplication of a vector by a number.
The special thing here is that (A . v), which is an n-vector,
in fact points in the same direction as v. If you pick
any vector w out of a hat, chances are that A.w and w are not
parallel.
There are loads of reasons why eigenvectors and eigenvalues
are interesting, but I guess that one of the main reasons
why geologists may be interested in them is that looking
at eigenvalues and eigenvectors gives a way of distilling
information out of a (typically big) matrix A. If you know
all eigenvectors and eigenvalues (there are n of them, with
an n-by-n square matrix), then you can reconstruct A completely;
however, if you only know, say, the 10 largest eigenvalues,
then---depending on your application---that may, in fact, be
all the information you really need. But, hey, this depends
on the application. Back to you for that part, Falk :-)
Mark
Oh yes, you asked for a reference. Any introductory book on
`linear algebra', and most introductory books on calculus, will
have a section on matrix analysis and eigenvalues/vectors.
PS if you want more explanation, or more discussion :-), email
directly to my email address.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|