With all my fancy book-learning and abstract axiomatic thinking, there are a lot of classical results that I’ve never learned. I just ran into an interesting classical theorem by Sylvester about symmetric matrices. Let An be the submatrix of the first n rows and columns. When you complete the square on the associated quadratic form, the coeffients of each square term is of the form
(det An)/(det An-1).
From this it follows that the matrix is positive definite if and only if each (det An) is positive.
The criterion is not practical for large matrices, but it does imply one interesting theoretical result: the set of positive definite matrices is a real semialgebraic set.