Find a unitary matrix whose first two columns are $$ \biggl [1/2, \iota/2, 1/2, \iota/2 \biggr ]^T \text{and} \biggl [\iota/2, 1/2, 1/2, -\iota/2 \biggr ]^T$$
I some how manage to find the other two columns by rearranging 1, $\iota$ and minus sign. What I obtain is
$$ \biggl [1/2, -\iota/2, \iota/2, 1/2 \biggr ]^T \text{and} \biggl [\iota/2, -1/2, -\iota/2, 1/2 \biggr ]^T$$
I tried to solve using the fact A$A^H$= I But it gets hairy. I want to know the correct way of doing this problem.