Representing a matrix as a product of two matrices

Question

I have a 4x4 matrix of complex numbers, say $X$ and I want to represent it as a product of a 4x1 & 1x4 vectors $x$ such that $$X = xx^H$$ where H denotes hermitian transpose of $x$. I am unsure how to proceed with this problem. Any hint/suggestion would be appreciated.

Answer 1

This is possible if and only if $X$ is a rank one Hermitian positive semi-definite matrix (or if $X$ is the zero matrix). If $X$ is of rank one, you can find a unit length column eigenvector $v \in \mathbb{C}^n$ of $X$ with $Xv = \lambda v$ and $\lambda > 0$. Then $X = (\sqrt{\lambda} v) \cdot (\sqrt{\lambda} v)^H$.