5

I have a unitary matrix that I want to construct. I only care what happens to the first computational state, so the first column is specified. So far, I've been assigning each question mark to a variable and solving $UU^T = I$ analytically. But this 6x6 case is out of computational reach for this method.

Is there any general method, or any clever trick, to help me fill in the rest of matrices such as $U$?

Note: I actually would prefer if all the entries were real, so technically these are better called orthogonal matrices.

$U = \frac{1}{\sqrt{5}}\begin{bmatrix} 0 & ? & ? & ? & ? & ? \\ 1 & ? & ? & ? & ? & ? \\ 1 & ? & ? & ? & ? & ? \\ 1 & ? & ? & ? & ? & ? \\ 1 & ? & ? & ? & ? & ? \\ 1 & ? & ? & ? & ? & ? \\ \end{bmatrix} $

Is there a general form of this method that you could post or link? – thespaceman – 2020-10-16T02:10:01.277

How are you wanting to generalise it? For an arbitrary first column, just follow exactly what I did. If it's more than one column you want to fix, you'd be better with a Gram-Schmidt procedure afaik. – DaftWullie – 2020-10-16T09:41:42.237