Suppose that $D$ is a division ring, $V$ is a finite-dimensional right vector space over $D$, and $f: V \to V$ is a $D$-linear map. Are there known canonical forms for $f$, in the spirit of the Jordan normal form or rational canonical form, but in the case where $D$ is noncommutative? I'm particularly interested in the case of $D = \mathbb{H}$ the quaternions, but would be very happy to see more general results too.
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You might want to check out
Cohn, P. (1995). Skew Fields: Theory of General Division Rings (Encyclopedia of Mathematics and its Applications). Cambridge: Cambridge University Press. doi:10.1017/CBO9781139087193
In particular they have chapters entitled things like
- Normal forms for matrix blocks over firs
- Normal forms for matrices over a tensor ring
- Normal forms for a single matrix over a skew field
rschwieb
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The reference given is incorrect: It is to the book review by Strambach, but the actual book is by P.M. Cohn alone and can be found at https://doi.org/10.1017/CBO9781139087193 – Bjorn Poonen Jun 19 '21 at 06:15
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