Building a Unitary Matrix from a Non-Unitary Matrix

[MrRobotoToo]

Suppose I have some arbitrary square matrix $M$, and I want to build a unitary matrix $U$: $$U=\left[\begin{array}{c|c}M & N \\\hline O & P\end{array}\right]$$ Does there exist some general procedure for determining $N$, $O$, and $P$ given $M$?

 

[Orodruin]

No, it may not be possible to do this depending on M.

 

[Orodruin]

Also, it depends on the size of your new matrix relative to $M$. Apart from the fact that you must ensure that $\sum_i |M_{ij}|^2$ and $\sum_j |M_{ij}|^2$. That being said, when you do have the possibility of constructing a matrix like that, I would first rewrite $M$ as $M_0 V$, where $V$ is a unitary matrix and $M_0$ is hermitian. Then you might get some inspiration from https://arxiv.org/abs/1107.3992