Editing Gram matrix (section)

===Positive-semidefiniteness===
The Gram matrix is [[symmetric matrix|symmetric]] in the case the inner product is real-valued; it is [[Hermitian matrix|Hermitian]] in the general, complex case by definition of an [[inner product]].

The Gram matrix is [[Positive-semidefinite matrix|positive semidefinite]], and every positive semidefinite matrix is the Gramian matrix for some set of vectors. The fact that the Gramian matrix is positive-semidefinite can be seen from the following simple derivation:
: <math>
  x^\dagger \mathbf{G} x =
  \sum_{i,j}x_i^* x_j\left\langle v_i, v_j \right\rangle =
  \sum_{i,j}\left\langle x_i v_i, x_j v_j \right\rangle =
  \biggl\langle \sum_i x_i v_i, \sum_j x_j v_j \biggr\rangle =
  \biggl\| \sum_i x_i v_i \biggr\|^2 \geq 0 .
</math>

The first equality follows from the definition of matrix multiplication, the second and third from the bi-linearity of the [[inner-product]], and the last from the positive definiteness of the inner product.
Note that this also shows that the Gramian matrix is positive definite if and only if the vectors <math> v_i </math> are linearly independent (that is, <math display="inline">\sum_i x_i v_i \neq 0</math> for all <math>x</math>).<ref name="HJ-7.2.10"/>