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The little bit of literature on infinite matrices I've been able to find studies a general setting in which the theory is hindered by constantly having to worry about whether or not various infinite sums converge or not. However, if we require all our matrices to have a only finite number of non-zero values on each row, a lot of these difficulties disappear (for example the product of two such matrices is always well defined and satisfies the same property on its rows). The problems I'm interested in only involve matrices of this type.

Is there any literature specifically about such matrices?

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    $\begingroup$ These are endomorphisms of a free module of infinite rank. $\endgroup$
    – Alex Degtyarev
    Commented Nov 14, 2014 at 23:42
  • $\begingroup$ What sort of properties of these matrices are you looking for? I would think that, despite perhaps having an infinite number of nonzero entries in some columns, these matrices would in many ways behave like finite dimensional matrices. Also, another interesting infinite matrix setting is upper-triangular matrices. If you simply require that all matrices are upper-triangular, you remove any convergence issues. $\endgroup$
    – Scott Andrews
    Commented Nov 15, 2014 at 0:01
  • $\begingroup$ Where $A$ is such a matrix, I'm interested in deducing as much information as I can about $A^n$, ideally getting a closed form for it. In particular, I'd love to know if there are analogues to diagonalization and similar methods. $\endgroup$
    – Jack M
    Commented Nov 15, 2014 at 0:07
  • $\begingroup$ @AlexDegtyarev: $\:$ That be for "a finite number of non-zero values on each" column. $\hspace{1.34 in}$ $\endgroup$
    – user5810
    Commented Nov 16, 2014 at 0:21
  • $\begingroup$ @RickyDemer Doesn't matter: just left vs. right modules. $\endgroup$
    – Alex Degtyarev
    Commented Nov 16, 2014 at 0:32

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