## How to force a matrix to be unitary given constraints on some of the elements?

0

I am working with a matrix of the following form:

$$A =\begin{pmatrix} a_{11} & Q & \ldots & Q\\ a_{21} & Q & \ldots & Q\\ \vdots & \vdots & \ddots & \vdots\\ a_{n1} & Q &\ldots & Q \end{pmatrix}$$

where the $$a_{ij}$$ elements are real and predetermined, the $$Q$$'s are placeholders and not necessarily equal to one another, and $$A$$ is square of size $$n$$x$$n$$. I am looking to find values for all $$Q$$'s such that $$A$$ is unitary. To do this I have attempted to set up a system of nonlinear equations of the form $$AA^\dagger=I$$ which yields a system of $$n^2-n$$ unknown $$Q$$'s, but only $$n^2/2 +n/2$$ equations after removing any duplicate equations. Therefore, for $$n=2$$ the system is over-determined and for $$n>3$$ the system is under-determined.

My question is, is there a method in which I can solve for $$Q$$'s to force $$A$$ to be unitary given these constraints for any size n?

Question was closed 2020-10-18T23:18:22.947

3

How about just performing the Gram-Schmidt Process?

Pick the other $$n-1$$ arbitrary linear independent vectors and perform Gram_schmidt process.

Caveat: This requires that the vector of the first column is already normalized. Otherwise we will have $$AA^* = cI$$ instead.

Update: I found another related/exact same question. Here is the link:

How can I fill a unitary knowing only its first column?

The issue that I found with this method is that once I apply the Gram-Schmidt process my column of $a_{11} ... a_{n1}$ is no longer the same. How do I apply this procedure while keeping the first column constant? – thespaceman – 2020-10-16T01:42:54.137

1Ah.. so you mean the vector $u_1 = \begin{pmatrix} a_{11} \ a_{21} \ \vdots \ a_{n1} \end{pmatrix}$ has a normalization factor in front of it now, right? – KAJ226 – 2020-10-16T02:35:10.357

1It is anyway a necessary condition that your first column is normalised. Otherwise $A$ can never be unitary. Picking random linearly independent vectors for the rest and performing Gram-Schmidt seems the optimal strategy to achieve your goal. Alternatively, you can directly pick random vectors from the orthocomplement of the preceding columns. This will not change the first column. – Markus Heinrich – 2020-10-16T10:33:01.123

The Gram-Schmidt method is exactly what I need when the first column is already normalized which is the case for me. – thespaceman – 2020-10-16T16:29:39.407