I showed that every Gram matrix, i.e. a $n \times n$ matrix $A$ with $A_{ij} = <x_i,x_j>$ 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 $< x_i,x_j> = A_{ij}$?

Any help is appreciated.

  • 3
    $\begingroup$ Every semi-definite matrix $A$ can be factored as $A=C^*C$ for some square matrix $C$. You can consider the columns of $C$ as your $x$'s. $\endgroup$ – Algebraic Pavel Jan 30 '14 at 1:38
  • $\begingroup$ Thank you so much! It helps a lot! $\endgroup$ – user112564 Jan 30 '14 at 5:29

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