# Solving for a Unitary Matrix

Hello everyone,

I'm trying to replicate nearly the following in Sage: https://mathematica.stackexchange.com...

The linked post is asking for $U$ such that $A = UB\overline{U}^{T}$ but I am looking for the following:

Given 2 lists of vectors say $A = ${$a_1, ..., a_n$}, $B =$ {$b_1, ..., b_n$}, is there a way to easily solve for a unitary matrix, $U$, such that {$a_1, ..., a_n$} = {$Ub_1, ..., Ub_n$}.

I understand I can do the following:

```
U = H.solve_left(W)
U /= U.norm()
U.is_unitary()
```

but it doesn't seem every efficient, as there may be other solutions that I cannot see using this method. I am new to Sage and any advice would be super helpful!

Thank you

Are you looking for any one solution or for all solutions?

I would be interested to see all possible solutions, but finding one Unitary Matrix satisfying the conditions would be enough for my purposes!

Are your "lists of vectors" form orthonormal bases by any chance?

Not necessarily, but from my list of vectors I guess I could extract a basis.