On Wed, 11 Jul 2001, George Hammond wrote:
> No, in general the Rank (R) of an NxN matrix is not the
> same as the Order of the matrix which is by definition, N.
> The number of evigenvalues and eigenvectors is equal
> to the Rank, not the Order. Naturally these R eigenvectors
> are non-zero or we wouldn't be talking about them. If you
> want to insist that there are actually N, but N-R of them
> are "zero eigenvectors", ok, but like I said I'm not here
> to jawbone about trivia.
George,
The rank of a matrix is the dimension of the space spanned by its columns
(or, equivalently, by its rows). In the special case of a diagonal matrix
this would be the number of non-zero columns (or rows), i. e. the number
of NON-ZERO eigenVALUES counted with multiplicity. Thus, for a
diagonalizable matrix (such as a real symmetric matrix) the rank is also
the number of non-zero eigenvalues counted with multiplicity.
You seem to have eigenvectors and eigenvalues confused in your discussion.
Gordon Brown
Department of Mathematics
University of Colorado
Boulder, CO 80309-0395
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