= Positive definiteness of a Gram matrix
{title2=$c^TGc=\|\sum_i c_ix_i\|^2$}
For real <vectors> $x_1,\ldots,x_n$ in a <Euclidean space>, their <Gram matrix> $G_{ij}=x_i^Tx_j$ satisfies
$$
c^TGc=\left\|\sum_i c_i x_i\right\|^2.
$$
Thus $G$ 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 $G_{ij}=\langle x_i,x_j\rangle$ gives
$$
c^*Gc=\left\|\sum_i c_i x_i\right\|^2.
$$
If the <inner product> is instead linear in its first argument, this definition of $G$ gives $c^*Gc=\|\sum_i\overline{c_i}x_i\|^2$. The <positive definiteness> criterion is unchanged, but the coefficient formula must respect the convention.
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