So now we're going to take one step
further towards a practical algorithm
for computing the eigenvector associated
with the smallest eigenvalue.  And how
does that work?  This is called the shift
of inverse power method. Okay? And before
we start discussing that in great detail,
let's do some homework.  The homework is
meant to give you insight into the
following question: If I know that lambda
and x are an eigenvalue and eigenvector
of A, can I somehow use that to come up
with convenient eigenvalues and
eigenvectors of the matrix A minus rho I
quantity inverse?  Okay?  This is known as a shifted matrix.  You're shifting A by rho.
But we're interested in here what
happens to the matrix when we invert it.
And you're also going to look at the
question of if I know X and Lambda such
that A times X equals X times Lambda,
Lambda being a diagonal matrix, can I
somehow gain some insight into the X and Lambda that come up when we diagonalize
the shifted matrix, or rather the
inverse of the shift of matrix.  So go
do that homework and then let's
reconvene.
