A real symmetric quadratic form $g(x)=\varSigma_{1\leq i,j \leq n}\,g_{ij} x^i x^j$ is positive (definite) if $g(x)>0$ for every $\mathbb{R}^n\ni x=(x^1,...,x^n)\neq 0$. It is well known that a real symmetric quadratic form is positive iff all eingenvalues of the symmetric matrix $M=(g_{ij})$ are positive. This can be verified applying Descartes' rule of alternating signs to the characteristic polynomial, as it has no complex roots.
Question: is there a considerably simpler way to recognize that $g(x)$ is positive without needing to find the characteristic polynomial of the associated matrix $(g_{ij})$?
This would be useful, for instance, to recognize that a family of maps $g_{ij}:\mathbb{R}^n\supset U\rightarrow\mathbb{R}$ defines locally a Riemannian metric $\langle v,w\rangle_y=\varSigma_{1\leq i,j \leq n}\,g_{ij}(y) v^i w^j$ on a manifold.