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Stein A. Strømme wrote:
>
> George Hammond <ghammond@mediaone.net> writes:
> ....
> > Next comes the interesting part. It is well known that all real
> > symmetric matrices possess eigenvalues and eigenvectors (see any math
> > text). Using large computers they can extract the eigenvalues and
> > eigenvectors from the 1000x1000 matrix. And, they can plot the
> > eigenvectors (which are vectors) as a graphical diagram. The first
> > thing they discovered is that there are only 13-eigenvectors. ...
>
> A 1000x1000 real symmetric matrix always admits 1000 linearly
> independent eigenvectors. This is the well known result that is
> referred to here.
[Hammond]
I hope you're not intent on playing mathematical trivia
with me.. I'm not interested in trivia. In general the
"rank" of a matrix is not the same thing as the "order"
of a matrix. An NxN matrix is by definition of order N,
however the rank of the matrix may be much less than N.
The rank of the matrix is the number of eigenvectors
(independent eigenvalues) of the matrix. Physically this
represents the dimensionality of the space described by
the matrix, which may be far less than the number, N, of
simple vectors in the space which make up the scalar
product matrix. I don't know where you got the idea that
order was equal to the rank unless you are talking about
matrices which have been reduced to "canonical form" or
something. It is a fact of life that "Factor Analysis"
is, above all else, all about determining the "rank" of
a correlation matrix (real symmetric matrix) where the
rank is known to be much smaller than the order of the
matrix. This is the entire rational for Factor Analysis;
that the rank is far less than the order. Being a
physicist, perhaps my mathematical terminology is not up
to snuff with the mathematics department or something?
>
> Stein
> --
> http://www.mi.uib.no/~stromme
-- Be sure to visit my website below, and please ask your news service provider to add alt.sci.proof-of-god ----------------------------------------------------------- George Hammond, M.S. Physics Email: ghammond@mediaone.net Website: http://people.ne.mediaone.net/ghammond/index.html -----------------------------------------------------------
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