Bilinear form/Symmetric/Definiteness/Minor criterion/Fact

Minor criterion for definiteness

Let $⟨ -, - ⟩$ be a symmetric bilinear form on a finite-dimensional real vector space, and let $v_{1}, \dots, v_{n}$ be a basis of $V$ . Let $G$ be the Gram matrix of $⟨ -, - ⟩$ with respect to this basis, and let $D_{k}$ denote the determinants of the square submatrices

M_{k} = {(⟨ v_{i}, v_{j} ⟩)}_{1 \leq i, j \leq k}, k = 1, \dots, n .

Then the following statements hold.

The form $⟨ -, - ⟩$ is positive definite if and only if all $D_{k}$ are positive.
The form $⟨ -, - ⟩$ is negative definite if and only if the sign in the sequence $D_{0} = 1, D_{1}, D_{2}, \dots, D_{n}$ changes at every step.