The oldest method for computing the
eigenvalues and eigenvectors of a
symmetric matrix, real-valued matrix, was
invented by Jacobi.  And I've actually
read the original paper regarding this
method.  It was written in German.  And
there's a little footnote there that
thanks his student Seidel for having
done all the calculations to a certain
number of digits of accuracy.  You can
imagine what that was like.  This was more than a hundred fifty years ago. Jacobi's
method also uses rotations but it uses
them a little bit differently.  Okay?  It
turns out that you can find a rotation,
Sigma -- sorry -- gamma, minus sigma, sigma,
gamma, transpose.  And then apply the same rotation from the other side
in such a way that the result of this is
a diagonal matrix.  Now obviously what
this does is it computes the Spectral
Decomposition of a two-by-two symmetric
matrix.  Okay? And we know that that has to be diagonal. Well, I have a little
homework where you try to figure out how
to compute these.  There is Jacobi's
method for computing the Spectral
Decomposition of a matrix.  And there is
Jacobi's methods for computing the
Singular Value Decomposition of a matrix.
And I'm going to describe the method for
the computing the Spectral Decomposition
first.  And then it will be pretty obvious
how we can then modify the algorithm in
order to compute the Singular Value
Decomposition.
