I showed that every Gram matrix, i.e. a $n \times n$ matrix $A$ with $A_{ij} = \langle x_i,x_j\rangle$ where $x_1,...,x_n$ are vectors in an inner product vector space $V$, is Hermitian and positive semi-definite.
But how to show the converse: For every Hermitian positive semi-definite matrix there is a inner product space $V$ and vectors $x_1,...,x_n$ such that $\langle x_i,x_j\rangle = A_{ij}$?
Any help is appreciated.