# Can infinite matrices represent nonlinear operators?

I read this reddit post and this SE thread discussing how to represent nonlinear/linear transforms in matrix notations but they were not sufficient.

In quantum mechanics, scientists use infinite matrices to represent operators. Should the operators be linear to be represented as matrices? If then, should the operators be linear even to be represented as infinite matrices? Or can infinite matrices represent nonlinear operators too?

• I'm not sure about nomenclature in context with "operators". But do you know the term "Carleman"-matrix (see wikipedia)? Or "Jabotinsky"-matrix? They are associated to functions like $\exp(x)$ etc and are of infinite size. But I don't know whether this is appropriately related to your "nonlinear-operator"-question... Aug 23 '18 at 9:54
• @GottfriedHelms I think that perfect. Thanks. Aug 24 '18 at 10:27
• @GottfriedHelms Well, where did you heard of that? I want to gather more information about that. Did you learn that while you were in university? Aug 25 '18 at 15:41
• I've learned about this in context of online-discussion about "tetration". Before that I've re-discovered that concept on my own by looking at (infinite) combinatorical matrices like Pascal-Matrix, Stirling-numbers-matrix and the like and experimenting with Neumann-series of such matrices. There is a couple of articles online freely accessible and something in "jstor". But I lack a rigorous introduction myself. So I tried to develop my knowledge by try&error. See for instance one attempt of an article: go.helms-net.de/math/tetdocs/ContinuousfunctionalIteration.pdf . But you might(...) Aug 25 '18 at 15:53
• (...) as well go to my math-index and look at the subdirectoried of combinatiorial matrices and of tetration. Most of it was developed and tested with use of that "Carleman"-matrix-concept Aug 25 '18 at 15:57

I'm reluctant to say it cannot be done, but I would think any natural use of matrices to represent functions would entail linear functions. For example, let $S$ be the set of convergent sequences of real numbers and let $A$ be an infinite matrix such that $X\in S\implies AX\in S$. That is $$(AX)_i=\sum_{j=1}^{\infty}a_{ij}x_j$$ where we know that sums will be convergent and the sequence of sums will converge. Then it follows from elementary properties of sums and sequences that $$A(aX+bY)=aAX+bAy$$