We now look at a useful theorem known as the Gershgorin  disk theorem.  And what
does it say?  Well, if you look at a matrix
-- and here we're looking at a 4 by 4
but obviously this generalizes -- and you
look at the diagonal elements of this
matrix and you place them in the complex
plane, then you can take the sum of the
absolute values of the off diagonal
elements in each row and consider those
to be a radius to a circle around these
points in the complex plane.  So, here we
have the radius rho_0, which is the sum
of the off diagonal elements, well the
absolute values of the off diagonal
elements in the row indexed with 0.  If you
now go to the point alpha_00 in the
complex plane, and you draw a circle
around that with radius rho_0 defined as
such.  And you do that for all of the
diagonal elements and the corresponding
radii -- so here we have another disk of
radius rho_1 and another disk of radius
rho_2 and another disk of radius rho_3.
Then what the theorem says is that all
the eigenvalues of the matrix must lie
in the union of these disks. Okay?  And that's particularly useful when for example you
know that these disks do not include the
origin, because then you know that the
matrix is non-singular.  Now proving this
is actually relatively straightforward
so you can just look at the proof in the
notes.
