So hopefully what you figured out is if
we take Ax equals lambda times x, then
we can always multiply both sides by A
inverse and we can divide both sides by
lambda. And if we then rearrange it what
we get is that A inverse times x is
equal to 1 over lambda times x.  Okay? So
what do we find out? if x is an
eigenvector of A, then x is an
eigenvector of A inverse.  And if lambda
is an eigenvalue of A then 1 over lambda
is an eigenvalue of A inverse.  And
obviously lambda cannot be 0.  But if A
is nonsingular then we know that none
of its eigenvalues are 0.  And therefore,
we know that 1 over lambda is
well-defined. Okay?  So that's the easy
part.
Now, what about this one?  Well, if we look at A inverse, that's the same as X lambda
X inverse inverse.  Well, that's a matter
of writing all of these in the opposite
order and then inverting each of them.
But inverting this, oops that should be
a lambda.  Inverting that gives you that.  We have a lambda inverse here and X
inverse here.  So what we notice is that
the same X that diagonalizes A is the
X that diagonalizes A inverse.  The lambda that shows up over here is
just the inverse of the lambda that showed up when we diagonalized A.  So
what do we notice? This is just equal to
the matrix that are the eigenvectors
of matrix A.  And then the diagonal matrix
that consists of the values one over
lambda_0 through one over lambda_m-1, and then the matrix X inverse.
Okay?  And from that we can totally read
off what we need.  If we have m linearly
independent eigenvectors for A, then we
have m in linearly independent
eigenvectors for A inverse.   And the diagonal that shows up over here is the inverse
of the diagonal that we got when we
diagonalized
A.  And therefore if we look at the
spectrum for matrix A, the spectrum for
matrix A inverse is just given by the
reciprocals of the eigenvalues of matrix
A.  Now importantly if lambda_0 was the
eigenvalue largest in magnitude of A
then it is 1 over lambda_m-1 that
is the eigenvalue largest in magnitude
for A inverse.  Okay?  So the eigenvector
associated with the largest eigenvalue
for A inverse is the eigenvector
associated with the smallest eigenvalue
of A.  And that gives us all the insight
we need to now transform the power
method into a method for finding the
eigenvector associated with the
smallest eigenvalue. Okay?
