  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.
