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Both matrix multiplication and quaternion multiplication are non-commutative; hence the use of terms like "premultiplication" and "postmultiplication". After encountering the concept of "quaternion matrices", I am a bit puzzled as to how one may multiply two of these things, since there are at least four ways to do this.

Some searching has netted this paper, but not having any access to it, I have no way towards enlightenment except to ask this question here.

If there are indeed these four ways to multiply quaternion matrices, how does one figure out which one to use in a situation, and what shorthand might be used to talk about a particular version of a multiplication?

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    One should, I think, talk about (left or right) H-module morphisms of free (left or right) H-modules, where H is the quaternions. There is an obvious notion of composition here.2010-08-09
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    Apologies, my grasp of the theory of modules is not up to snuff; care to expound to a non-specialist?2010-08-09
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    I'm not confident that the above agrees with the standard definition. The standard definition, as far as I can tell, is to take the product in the "obvious" order, e.g. in the terms of the product AB the component of A always appears before the component of B.2010-08-09
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    When you say quaternion matrices are you refer to representing quaternions as matrices as mentioned here http://en.wikipedia.org/wiki/Quaternion#Matrix_representations or something else entirely?2010-08-09
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    I believe he is referring to matrices whose entries are quaternions. (Using the standard representation of quaternions as matrices, one can turn these into complex matrices.)2010-08-09
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    Qiaochu nailed it; I am already aware of the equivalence of quaternion algebra and the algebra of 2-by-2 complex/4-by-4 real matrices. That now presents the equivalent problem of a "matrix of matrices".2010-08-09
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    Cool, would an example problem be a situation where you were trying to multiply 2 quaternions by a single quaternion, and thus have a real 4x4 times a 4x8, or are you also interest in more general situations beyond just augmenting one quaternion onto another?2010-08-09
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    My motivation is a bit more theoretical than my usual fare: I am figuring out how to consistently extend the standard methods of numerical linear algebra (LU, QR, SVD, etc.) to quaternion matrices. Since all of these have matrix multiplication as a necessary building block (let me worry about how to consistently divide quaternions later :P ), I want to figure out how pre- and post-multiplication by matrices should work out if the matrices have quaternion elements. I have seen some literature on quaternion matrix methods, but wish for a more organic starting point.2010-08-09
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    Gotcha, my experience with quaternions has been not on the theory side at all. I thought you might interested in implementation for numerical calculations, but if your not, I won't be of much help. It sounds interesting though.2010-08-09
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    Well of course once I figure out how to make this whole multiplication business consistent, I will be eventually implementing these algorithms. What's an algorithm without actual working code? ;)2010-08-09

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