Check for positive definiteness or positive semidefiniteness
I assume you already know your matrix is symmetric. 
A good test for positive definiteness (actually the standard one !) is to try to compute its Cholesky factorization. It succeeds iff your matrix is positive definite.
This is the most direct way, since it needs O(n^3) operations (with a small constant), and you would need at least n matrix-vector multiplications to test "directly".
I want to check if a matrix is positive definite or positive semidefinite using Python.
How can I do that? Is there a dedicated function in SciPy for that or in other modules?