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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](http://en.wikipedia.org/wiki/Gram–Schmidt_process)2012-04-08
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    @t.b. I have tried Gram-Schmidt but failed. I don't know how to choose other two vectors to apply Gram-Schmidt.2012-04-08
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    @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...2012-04-08

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