Sylvester's Minorant Criterion, Lagrange-Beltrami Identity, and Nonnegative Definiteness

TL;DR

This paper reviews classical criteria for positive and nonnegative definiteness of quadratic forms, providing elementary proofs for forms up to three variables and deriving explicit identities for ternary cases.

Contribution

It offers an elementary, self-contained proof of Sylvester's Criterion for small quadratic forms and derives explicit Lagrange-Beltrami identities for ternary forms.

Findings

Elementary proof for quadratic forms up to 3 variables

Explicit Lagrange-Beltrami identity for ternary forms

Characterizations of positive and nonnegative definiteness

Abstract

We consider the characterizations of positive definite as well as nonnegative definite quadratic forms in terms of the principal minors of the associated symmetric matrix. We briefly review some of the known proofs, including a classical approach via the Lagrange-Beltrami identity. For quadratic forms in up to 3 variables, we give an elementary and self-contained proof of Sylvester's Criterion for positive definiteness as well as for nonnegative definiteness. In the process, we obtain an explicit version of Lagrange-Beltrami identity for ternary quadratic forms.