I have a matrix $F ∈ \mathbb{C}^{(m × N)}$, where $N > m$, and $F \times F^H$ is a unitary $m × m$ matrix.
I need to find a unitary matrix $G$ with a dimension of $N × N$ such as results of $F\times G=0$ with dimension $m \times m$ or at least the result is minimum.
So, does the matrix $G$ can be found directly or it must to be optimized? If optimized, can I proof that is always exist?