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

Question

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?

Answer 1

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.

Answer 2

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...

Answer 3

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 ...