 Positive definiteness of a Gram matrix

For real vectors $x_1,\dots ,x_n$ in a Euclidean space, their Gram matrix $G_{ij}=x_i^T{x}_j$ satisfiesThus $G$ is a positive semidefinite matrix. It is a positive-definite matrix exactly when the vectors are linearly independent: the squared norm vanishes precisely when the indicated linear combination vanishes.

For complex vectors, specify the convention. With the inner product conjugate-linear in its first argument, the same definition $G_{ij}=⟨x_i,x_j⟩$ givesIf the inner product is instead linear in its first argument, this definition of $G$ gives $c^{\ast }Gc=\Vert {\sum }_i\overline{c_i}{x}_i{\Vert }^2$. The positive definiteness criterion is unchanged, but the coefficient formula must respect the convention.