# Name for a matrix similar to a companion matrix? Translating "Monogène" matrix from French

I am currently translating a research paper from French (which I do not speak well). I have made good progress with copious use of google translate & switching between French & English versions of articles on wikipedia, coupled with knowledge in the given field. However, I am stuck on the following ($M$ is a matrix):

The parenthetical implies that $M$ is similar to a Companion Matrix, but I am stumped when it comes to what "monogène" means. Google translate says "monogenic", which I would infer to be the author's term for "similar to the companion matrix of a polynomial." However, I can't say that I've come across the term "Monogenic Matrix" before, plus this would just be the Froebius normal form / rational canonical form of a matrix with only one block.

Is this a reasonable translation/interpretation?

• I've never seen this term, but it evidently refers to the fact that the action of $M$ on the $K$-vector space $V$ makes $V$ into a $K[X]$-module generated by a single element. I would suggest changing your title to something like "name for a matrix similar to a companion matrix," since this isn't really about French. Commented Feb 24, 2017 at 7:46
• Okay, that seems reasonable from the context. Felt weird thinking of a single matrix (rather than group / field) as monogenic, wanted to make sure I wasn't missing something Commented Feb 24, 2017 at 7:52

Such a matrix is said to be non-derogatory in English. The condition is equivalent to one of

• having equal minimal and characteristic polynomials,

• having a cyclic vector,

• being similar to a companion matrix.

• Aha! That's the magic word. Commented Feb 24, 2017 at 7:59
• I've head non-derogatoire used in French, too, by the way. Commented Feb 24, 2017 at 8:01

I'm in France since 20 year but this is the first time that I hear this term. For curiosity, I Googled a little bit and found that this term is referred as a synonym of 'cyclic' in group theory. In the sense that a group with a generator is called a monogène group. This matches with the Latin meaning of the word: mono-genesis ie. generation by a single element.

Hope this helps in clarifying...

• Thank you! This plus the comment by user49640 clears it up. Commented Feb 24, 2017 at 7:53

Not an answer ... just trying to explain a bit more ...

Consider a finite dimensional vector space $E$ and an endomorphism $u$ of $E$.

In french, it is common to call $u$ un endomorphisme cyclique iff there exists $a\in E$ such that $(a,u(a),\cdots,u^{n-1}(a))$ is a basis of $E$.

It can be shown that this condition is equivalent to the existence of a basis $\beta$ of $E$ in which $u$ is represented in $\beta$ by a companion matrix.

So any square matrix which is similar to a companion matrix could be called a "cyclic matrix" or maybe a "monogenous matrix."

I don't know if it is somehow an official name, though ...