Find unitary matrix whose first two columns are given.

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.

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Gram-Schmidt – t.b. Apr 8 '12 at 18:44
@t.b. I have tried Gram-Schmidt but failed. I don't know how to choose other two vectors to apply Gram-Schmidt. – Faisal Apr 8 '12 at 18:47
@Faisal: it doesn't matter which vectors you choose unless you are unlucky and they are part of the vectorspace spanned by the vectors given... – Fabian Apr 8 '12 at 19:02

Given vectors $v_1,\dots,v_k$ with $k<n$, you can add all the standard basis vectors $e_1,\dots,e_n$ to the list and apply the Gram-Schmidt algorithm to $v_1,\dots,v_k,e_1,\dots,e_n$. If a vector becomes zero after the subtraction step, you throw it away. Since the algorithm preserves the linear span of the collection of vectors, you are guaranteed to end up with a basic of the space.