Generating complex unitary matrix with a given column

Question

I'm looking for a way to generate a complex unitary matrix in which the last column is given. I will appreciate any tips and hints.

Thanks!

Answer 1

The Graham-Schmidt process can be used to orthonormalize a set of linearly independent vectors in any inner product space. I assume the given vector $v$ is a unit vector, so not all its components are $0$. Say entry $k\neq 0$. Then make the fisrt $n-1$ columns equal to $$e_1,e_2,\dots,e_{k-1},e_{k+1},\dots,e_n$$ where $_k$ is the $k$th natural basis vector, so that the columns are linearly independent, and apply Graham-Schmidt.