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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?

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    $\begingroup$ 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... $\endgroup$ Aug 23 '18 at 9:54
  • $\begingroup$ @GottfriedHelms I think that perfect. Thanks. $\endgroup$ Aug 24 '18 at 10:27
  • $\begingroup$ @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? $\endgroup$ Aug 25 '18 at 15:41
  • $\begingroup$ 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(...) $\endgroup$ Aug 25 '18 at 15:53
  • $\begingroup$ (...) 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 $\endgroup$ Aug 25 '18 at 15:57
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In some sense everything you can possibly define can be encoded in numbers and you are free to write those numbers in matrix form to get a linear operator. Thus the answer to your question is trivially yes.

However, the answer is also nontrovially yes. Such a yes is dependent on what connections between the linear and the nonlinear operations you consider meaningful. One way to make such a meaningful connection is the use of Koopman operators in dynamical systems https://faculty.missouri.edu/~liyan/Coll.pdf

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  • $\begingroup$ Thanks, but unfortunately I cannot open the link.. $\endgroup$ Sep 1 '18 at 13:26
  • $\begingroup$ Nope, it opens well in chrome browser. $\endgroup$ Sep 1 '18 at 13:30
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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 $$A(aX+bY)=aAX+bAy$$

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  • $\begingroup$ So you mean it's impossible with normal operations. $\endgroup$ Aug 26 '18 at 4:11
  • $\begingroup$ As I said, I'm reluctant to use a word like "impossible," and I don't know exactly what you mean by "represent." In the example of Carleman matrices that Gottfried gave, you certainly have a matrix associated with a nonlinear function, but I don't know that I'd say the matrix "represents" the function. $\endgroup$
    – saulspatz
    Aug 26 '18 at 9:19

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