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Suppose $M \leq N$ are positive integers and let $r_1,\ldots,r_M$ represent the rows of an $M\times N$ matrix $A$ in which: (i) the rows of $A$ are orthogonal and having the same norm, (ii) the columns of $A$ all have equal norm, and (iii) the entries of $r_i$ are all rational numbers. My question is when is it possible to extend this to an orthogonal-like $N \times N$ matrix with rational entries? That is an $N \times N$ matrix $Q$ so that $QQ^{T} = c\cdot I$ for some constant $c$.

I'm looking for any references possible that might lead to the answer. It's an interesting question, which I think might not be too obvious and may even be unknown? Thanks!

Suugaku
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    If the columns have equal norm that is a perfect rational square, or can be written as the sum of 2 rational squares, it can be done. – Calvin Lin May 18 '13 at 00:43
  • Can you elaborate on that? I'd appreciate it! – Suugaku May 18 '13 at 00:46
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    Witt's Theorem does the perfect square (i.e. norm 1) case. – Calvin Lin May 18 '13 at 00:48
  • Awesome, I'll look into it. I thought this might drastically relate to rational quadratic forms, but unfortunately my number theory is not so good. – Suugaku May 18 '13 at 00:51
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    See here for a non-quadratic formulation writeup. Though, those ideas are underlying it. – Calvin Lin May 18 '13 at 00:53
  • That's really awesome! Thanks!! – Suugaku May 18 '13 at 00:59
  • Can I ask if you have a reference to the two rational squares case, or does it follow from the one rational square case? – Suugaku May 18 '13 at 01:14
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    It's similar. Instead of sending 1 vector to $Ne_1$, you want to send 2 vectors to $Ae_1 + Be_2$ and $Be_2 - A e_1$. – Calvin Lin May 18 '13 at 01:21
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    See related http://math.stackexchange.com/questions/261204/every-integer-vector-in-mathbb-rn-with-integer-length-is-part-of-an-orthogon where the complete question was answered by J.S.Hsia in 1978: Two theorems on integral matrices. Linear Multilinear Algebra 5, 257-264 (1978). – Will Jagy May 18 '13 at 02:57

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