Let's have a look at what happens
to this matrix right here.  Alright?
If everything is converging correctly,
then what we expect to happen to this
matrix is that this last entry right here
ends up converging to the eigenvalue
that is smallest in magnitude.
All of these values end up converging to
zeros.
And all of these values by symmetry
will also converge to zeros.
Now we're going to do an exercise
that shows that if we have a block
diagonal matrix that is in this particular case Hermitian --
but actually that's a minor point --
then the eigenvalues and eigenvectors of
the two blocks on the diagonal can be
computed separately and easily integrated
into eigenvalues and eigenvectors
of the whole matrix.  What that means
is that once these values become
sufficiently small,
then for practical purposes the matrix
is blocked diagonal.
It's actually even simpler than that.
We've just peeled off
one entry at the end of the diagonal.
And our algorithm can then what is
called deflate. And we can continue our algorithm by
focusing on this submatrix.  And we can then pick
our shift to always be the bottom right
entry
of that submatrix.  And in that way
we will eventually deflate down to a one
by one matrix, at which point we have found all of our
eigenvalues and eigenvectors.  Now because while we are actually
converging to this eigenvalue at a rate
that is similar to the Rayleigh Quotient
Iteration -- which is very quick
indeed -- we are also applying
the Power Method to the rest of the
matrix.  It is entirely possible that we are
fortunate that along the way somewhere zeros
appear in such a way that we can actually
deflate into smaller subproblems that are still
a little bit more substantial.  And if
that happens, that's just
a happy additional benefit of the
algorithm.
