= Positive definiteness of a Gram matrix
{title2=$c^TGc=\|\sum_i c_ix_i\|^2$}
For real <vectors> $x_i$ in a Euclidean <inner product space>, their <Gram matrix> $G_{ij}=x_i^Tx_j$ 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 $c^*Gc=\|\sum_i c_ix_i\|^2$.
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