Simple Theorem on Hermitian Matrices and an Application to the Polarization of Vector Particles

It is proven that a necessary and sufficient condition for an $n$-dimensional Hermitian matrix $\rho$ to be positive definite is that it be expressible in the form $\rho =OEO†$, where $O$ is a complex orthogonal matrix and $E$ is a diagonal matrix with positive elements. This accomplishes a parametrization since $O$ has $n^2-n$ real parameters and $E$ has $n$ of them. The proof is constructive, giving $O$ and $E$. It is further shown that the limit forms of this expression yield all the non-negative definite matrices. The parametrization for the polarization matrix of a spin-one particle is given explicitly.