On increasing quaternion matrices

Both matrix reproduction and also quaternion reproduction are non-commutative; therefore making use of terms like "premultiplication" and also "postmultiplication". After running into the principle of "quaternion matrices", I am a little bit puzzled regarding just how one might increase 2 of these points, given that there go to the very least 4 means to do this.

Some looking has actually netted this paper, yet not having any kind of accessibility to it, I have no other way in the direction of knowledge other than to ask this inquiry below.

If there are without a doubt these 4 means to increase quaternion matrices, just how does one identify which one to make use of in a scenario, and also what shorthand could be made use of to speak about a certain version of a reproduction?

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2019-05-06 21:43:35
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Answers: 3

I presume I need to expand my comment right into a solution. Offered 2 matrices $a_{ij}$ and also $b_{ij}$ with access in any kind of (associative) ring $R$, the all-natural definition of the item has access

$\displaystyle c_{ij} = \sum_k a_{ik} b_{kj}.$

This reproduction is associative, and also it additionally concurs with the reproduction one gets from any kind of limited - dimensional matrix depiction of $R$ by changing each access by the equivalent matrix.

I do not see any kind of certain factor to take into consideration a various idea of reproduction. Scuffing of several of the reproductions appears ridiculous to me, and also increasing in the contrary order offers you basically the very same reproduction.

This definition does not concur with the definition in my first comment ; reproduction by among the above matrices does not specify an $R$ - component homomorphism when $R$ is noncommutative.

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2019-05-10 08:53:14
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This offers fairly an instinctive suggestion concerning what is taking place :
http://plus.maths.org/content/curious-quaternions

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2019-05-09 10:58:30
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... however I am just permitted to upload one link per solution - so below is the follow up http://plus.maths.org/content/ubiquitous-octonions

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2019-05-09 10:58:11
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