Timeline for Centralizer of a Matrix over a Finite Field
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Aug 20, 2012 at 11:34 | comment | added | Jim Humphreys | @A Stasinski: Yes, it should be just "regular" here (in Steinberg's sense), e.g., for semisimple matrices all eigenvalues are distinct and for nilpotent matrices the Jordan normal form has a single block. A "regular" matrix over an algebraically closed field is one for which the centralizer has smallest possible dimension, here n. | |
| Aug 20, 2012 at 10:12 | comment | added | A Stasinski | @Jim Humphreys: In this situation "cyclic" is the same as "regular" since any companion matrix is cyclic. In the case of $GL_n$-conjugacy the term cyclic actually suggests more than regular because it comes from the fact that $F^n$ is a cyclic $F[A]$-module. Still I prefer the term "regular" because it highlights the connection to the more general theory. | |
| Aug 19, 2012 at 22:42 | comment | added | Jim Humphreys | Terminology is always a problem in matrix theory, but "cyclic" matrix doesn't suggest much. At least from the viewpoint of Jordan decomposition in linear algebraic groups or their Lie algebras, the current term for such a matrix is "regular semisimple". I recall also an archaic matrix term "nonderogatory" in a similar framework. In any case, Jordan decomposition is a powerful tool to organize centralizers and dimensions even in this classical matrix algebra situation, adapted to Lie algebras over finite fields. | |
| Aug 19, 2012 at 20:49 | vote | accept | zacarias | ||
| Aug 19, 2012 at 20:27 | history | answered | Alireza Abdollahi | CC BY-SA 3.0 |