Prove that a certain matrix exists

Question

I need to prove that, for every metric tensor $G$, there exists a matrix $C$ such that $G = C^T*C$.

Answer 1

$G$ is symmetric bilinear form and so is represented by a symmetric positive definite matrix (I suppose you call that matrix $G$ too). That matrix has all eigenvalues positive (positive definite) and satisfies $G=H^TDH$ where $H$ is an orthogonal matrix and $D$ is diagonal with (positive!) eigenvalues on the diagonal. Take $Q=\sqrt{D}$ the square root of $D$ (which is a diagonal matrix with the values on the diagonal equal to square roots of the corresponding values on $D$, $Q^TQ=Q^2=D$) and then notice that $G=H^TQ^TQH$, i.e. you can take $C=QH$.