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Problem #17 in Zhan's survey of open problems in matrix theory is the Li-Poon problem on writing a square real matrix as the linear combination of $k$ orthogonal matrices. They proved that it is possible to take $k=4$ for every square real matrix and asked if $k$ is the least possible such number.

I am wondering if there is a ring-theoretical version of this conjecture. This raises some even more basic questions:

What is the ring-theoretic equivalent of an orthogonal matrix?

And since an orthogonal matrix $O$ can be characterized as satisfying $OO^{T}=I$,

What is the ring-theoretic equivalent of the matrix transpose operation?

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What do you mean by ring-theoretic? Replace the ground field by a ring, or replace the matrix ring by a ring? Is the ring commutative or noncommutative? –  darij grinberg Feb 9 '13 at 0:09
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Probably too general, but a starting point could be a ring $R$ with an anti-automorphism $t:R\to R$ of order $2$. Then $O(R):=\{x\in U(R)|t(x)=x^{-1}\}$, where $U(R)$ denotes the units of $R$. –  Julian Kuelshammer Feb 9 '13 at 0:22
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For the ring version of "transpose", I think the story begins here: en.wikipedia.org/wiki*-algebra –  Suvrit Feb 9 '13 at 0:45
    
(the link is broken, but it points to the wiki page for 'star algebra' ) –  Suvrit Feb 9 '13 at 0:46
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In the linked survey of Zhan, the result mentioned is a linear combination of 4 orthogonal matrices. Along the same lines, I (with David Radcliffe) discovered that for all orders n and many rings R (including fields), with finitely many exceptions every matrix is the sum of exactly two invertible matrices. If I find the MathOverflow question related to this, I will return with a link. Gerhard "Ask Me About Matrix Sums" Paseman, 2013.02.08 –  Gerhard Paseman Feb 9 '13 at 4:52

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