 Positive definiteness of a Gram matrix

For real vectors $x_i$ in a Euclidean inner product space, their Gram matrix $G_{ij}=x_i^T{x}_j$ satisfies the displayed identity by expanding the squared norm. Thus every Gram matrix is a positive semidefinite matrix, and it is a positive-definite matrix exactly when the vector family is linearly independent. The complex version replaces transposes by conjugate transposes and uses $c^{\ast }Gc=\Vert {\sum }_i{c}_i{x}_i{\Vert }^2$.