Now there's actually a stronger result.
And it goes as follows: Let's take our
matrix A and let's view it as its
diagonal, which is where these points in
the complex plane where the disks are
centered are, plus its off diagonal part
and then let's parameterize this by
introducing a parameter, omega.  So now we
have a family parameterized with omega,
that ranges from when omega is equal to
zero it is purely the diagonal part of A
and when omega is equal to 1 you end up
with the matrix A.  Okay? Now if you think
about it, when omega is equal to zero,
this is just the diagonal matrix.  And
what we know is that its eigenvalues are
just the values on its diagonal.  And you
can think of the Gershgorin disk theorem
saying an eigenvalue lies in the disk of
radius zero.  Well all of the eigenvalues
lie in the union of the disks of radius
zero around those points.  But we actually
know that those points are the
eigenvalues of our matrix D of our
matrix A of zero.  Okay?  Then we can start varying our omega.  And an insight is that
where the eigenvalues of matrix A are
varies continuously with this parameter
omega. And that's you know that's another argument.  But as far as the Gershgorin
disk theorem is concerned, what this
means is that when omega is very small
these disks are very small.  And because
of this continuity argument what we know
is that only eigenvalues that were in
the disk
when the disks were smaller are now still in the disk, as long as these disks
stay disjoint from each other.  And it's
only when these disks start overlapping
that an eigenvalue from one disk could
wonder into the other disks.  Well what
this means is that when finally omega is
equal to 1, you're back to the conditions
of the Gershgorin disk theorem.  But you
what you now have observed is that the
number of eigenvalues that are in the
union of a set of overlapping disks is
exactly equal to the number of those
overlapping disks when those overlapping
disks are disjoint from all of the other
disks.  And that allows you to better
qualify and quantify where the
eigenvalues might be.
