I Building a Unitary Matrix from a Non-Unitary Matrix

1. May 19, 2017

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 $M$?

2. May 20, 2017

Orodruin

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

3. May 20, 2017

Orodruin

Staff Emeritus
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