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?
Best Answer
I guess I should expand my comment into an answer. Given two matrices $a_{ij}$ and $b_{ij}$ with entries in any (associative) ring $R$, the natural definition of the product has entries
$\displaystyle c_{ij} = \sum_k a_{ik} b_{kj}.$
This multiplication is associative, and it also agrees with the multiplication one obtains from any finite-dimensional matrix representation of $R$ by replacing each entry by the corresponding matrix.
I do not see any particular reason to consider a different notion of multiplication. Changing the order of some of the multiplications seems nonsensical to me, and multiplying in the opposite order gives you essentially the same multiplication.
This definition does not agree with the definition in my first comment; multiplication by one of the above matrices does not define an $R$-module homomorphism when $R$ is noncommutative.