# A name for matrices with only simple eigenvalues?

I am constantly working with hermitian matrices without multiplicity in their spectrum. Since this hypothesis appear in several important problems, for instance perturbation theory, I looked in the literature for an accepted terminology but found nothing. Does anyone know a reference where these matrices, or their set, have been given a name ? I am considering calling them "simple matrices" but it is a bit ambiguous...

• I don't know of any established terminology but surely `simple' is a candidate for the most over-used adjective in mathematics. Anything else would be better!! – Nick Gill Oct 18 '13 at 7:31
• @NickGill: Anything? Let's try out randomlists.com/random-adjectives to test that. "Paltry matrices" or "upbeat matrices", perhaps? – Mark Meckes Oct 18 '13 at 8:02
• «Multiplicity-free» is a good name. – Mariano Suárez-Álvarez Oct 18 '13 at 8:03
• Another suggestion is separable, because the characteristic polynomial is separable. – Peter Mueller Oct 18 '13 at 11:25
• To me a simple matrix is one whose characteristic polynomial is irreducible. – Amritanshu Prasad Oct 18 '13 at 11:40

## 4 Answers

Since your matrices are Hermitian, this is the same as non-derogatory ... see http://en.wikipedia.org/wiki/List_of_matrices for terminology derogatory matrix.

In the context of algebraic groups, these are the regular semisimple elements of $GL_n$. The "semisimple" part means diagonalizable, and the "regular" part means that the centralizer has dimension $n$. So you could call them regular semisimple.

• I like this answer, but a quick search on google showed that "non derogatory" is a much more common name in linear algebra (altough I never heard of it before). So I accepted Gerald's answer. – Fabien Besnard Oct 19 '13 at 7:17
• Sure ! I gave you the terminology that people who work with algebraic groups use, but you should choose the one that will sound familiar to the people who read you ! – Aurel Oct 19 '13 at 9:41

Apparently, there is something I don't understand in this discussion. Why should one reinvent the wheel? Such matrices have always been known as Hermitian matrices with simple spectrum. Just look at this wiktionary entry or Terry Tao's blog out of thousands of other examples.

Since "almost all" (in the Lebesgue sense) matrices have multiplicity-free spectrum, you can use something like "general position" or "generic".

Alternatively, "multiplicity-free spectrum matrix" or "MFSM" could also work.