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Is there a matrix way of writing system of linear equation over a skew field where the variables in the equations are both left multiplied and right multiplied by elements of the skew fields. Is the complexity of solving such equations ( say over the quaternions) known.

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If your skew field is a finite dimensional algebra over a commutative field (as e.g. the quaternions) then using a basis will quickly reduce your question to standard linear algebra. –  Felipe Voloch Oct 27 '11 at 14:18
    
Almost everything known about linear equations over general skew fields can be found in the book by P. M. Cohn, "Skew Fields", Cambridge University Press, 1995. –  Anatoly Kochubei Oct 27 '11 at 15:27
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