This concept addresses how to determine, for a symmetric matrix depending on a parameter, the values for which it is positive definite versus positive semi-definite. It presents three equivalent theoretical tests from linear algebra — the leading principal minors (determinants) test, the pivots test, and the quadratic-form/completing-the-square test — each grounded in the formal definition of positive (semi-)definiteness via the sign of the associated quadratic form. The topic sits within the broader theory of quadratic forms and symmetric matrices, linking determinants, elimination pivots, and eigen-structure as equivalent characterizations of definiteness.
This concept addresses how to determine, for a symmetric matrix depending on a parameter, the values for which it is positive definite versus positive semi-definite. It presents three equivalent theo…