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

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?

## 1 Answer

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? Oct 16 '20 at 1:42
• Ah.. 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? Oct 16 '20 at 2:35
• It 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. Oct 16 '20 at 10:33
• The Gram-Schmidt method is exactly what I need when the first column is already normalized which is the case for me. Oct 16 '20 at 16:29