Let , and assume that is positive definite, is positive semidefinite whence exists (where denotes transpose).

In what sense does minimize the quadratic expression , over ?

Is this minimization of over all vectors with respect to the usual ordering for positive semidefinite matrices iff is positive semidefinite?

Next consider the extension , where and diagonal, where all other matrices are as above.

What is the minimum over (with respect to the previous ordering or something )??

Any help will be deeply appreciated.