A classic result in theoretical linear
algebra is given any matrix A, it may
or may not be the case that is
diagonalizable, but what is the case is
that you can find a matrix X such that X
inverse A times X equals a matrix that
is block diagonal.  Each of these blocks
on the diagonal is a Jordan block.  So it
has this form.  It could be a one by one
Jordan block.  If all of these Jordan
blocks are one by one, then the matrix in
fact is diagonalizable.  But in general, it
may be the case that more or even all of
these blocks are proper Jordan blocks
that are not one by one. The diagonal of
these Jordan blocks then are
eigenvalues of this matrix. It's entirely
possible that this eigenvalue is equal
to that eigenvalue. The total number of
times that an eigenvalue appears along
the diagonal of this matrix is then the
algebraic multiplicity of the eigenvalue.
It's the multiplicity that that eigenvalue
is a root of the determinant.  The geometric multiplicity of an eigenvalue
equals the number of Jordan blocks that
have that eigenvalue as its eigenvalue.
In other words, it's equal to the number
of linearly independent eigenvectors
associated with that eigenvalue.  Now
this is a very nice result. One can prove
its existence. We can spend a lot of time
on this, but the fact is that it has no
real practical value. I know of no practical linear algebra
software library that even provides a
routine for computing it. So it's a nice
theoretical result.  It allows us to
discuss things like the algebraic and
geometric multiplicity of an eigenvalue.
But we're not going to spend any more
time on it because we're interested in
how we actually take theory and make it
practical.
