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We consider the characterizations of positive definite as well as nonnegativedefinite quadratic forms in terms of the principal minors of the associatedsymmetric matrix. We briefly review some of the known proofs, including aclassical approach via the Lagrange-Beltrami identity. For quadratic forms inup to 3 variables, we give an elementary and self-contained proof ofSylvester's Criterion for positive definiteness as well as for nonnegativedefiniteness. In the process, we obtain an explicit version ofLagrange-Beltrami identity for ternary quadratic forms.

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