Re: [math-fun] Positive eigenvalues imply positive definite?

Dan Asimov (hi Dan!) writes: A nice condition equivalent to positive definiteness is that all the

upper-left

k x k submatrices have positive determinant.

Why is this true? If it's true, it has the non-obvious consequence that "all upper-left submatrices have positive determinant" is equivalent to "all lower-right submatrices have positive determinant", right? Jim Propp