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For example, the matrix could have finitely many rows and columns, but each row/column has uncountably many elements and you can do the standard matrix multiplication by taking care to match up the entries with corresponding pairs of real number indices.

Do such objects exist and has there been any work on them?

Does

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    $\begingroup$ "finitely many rows and columns, but each row/column has uncountably many elements" What does that mean? Can you give an example? $\endgroup$ – Chris Culter Nov 10 '17 at 19:41
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    $\begingroup$ If there are finitely many rows and columns then number of elements is just rows times columns, no ? $\endgroup$ – A---B Nov 10 '17 at 19:42
  • $\begingroup$ Do you mean that each element in the matrix is some uncountable set? $\endgroup$ – Emil Nov 10 '17 at 19:47
  • $\begingroup$ Are you expecting your matrices to represent linear transformations between infinite-dimensional vector spaces? $\endgroup$ – Rob Arthan Nov 10 '17 at 19:50
  • $\begingroup$ @RobArthan not really. I'm more interested in their properties as a ring/module $\endgroup$ – Zachary F Nov 10 '17 at 21:36
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Two such generalizations come to mind: integral operators defined by a "kernel" $T(f)(x) = \int K(x, y)\ f(y) dy$. Such operators compose by convolving kernels $\int K_1(x, y) K_2(y, z) dy$, which is evidently a continuous generalization of matrix multiplication. The second generalization is less obviously a direct generalization, but here it is: elements in an arbitrary von Neumann factor of type II$_1$ can be regarded as continuous generalizations of finite matrices.

But you should abandon the idea of "finitely many rows and columns" and think "continuously indexed rows and columns" instead.

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  • $\begingroup$ Yes, continuously indexed is what I was initially thinking. I got confused by the multiplication thing while I was typing the question. $\endgroup$ – Zachary F Nov 10 '17 at 19:52
  • $\begingroup$ I think the confused description in the question could be interpreted as a matrix of continuously indexed matricies $\endgroup$ – Zachary F Nov 10 '17 at 21:39
  • $\begingroup$ Or perhaps a four dimensional matrix with two of the dimensions continuously indexed and the other two finite $\endgroup$ – Zachary F Nov 10 '17 at 21:41
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Infinite matrix rings are pretty interesting. I know several interesting examples.

The simplest is the ring of column-finite matrices over a field $F$, which is isomorphic to the ring of linear transformations of an infinite dimensional $F$ vector space. here are some of its properties. There is also, of course, the ring of row-finite matrices, and the row-and-column finite matrix rings. You could also consider the matrices with finitely many nonzero entries, and generate the ring containing those and the identity matrix.

There is also a close-knit cluster of three examples due to Bergman that all live inside infinite matrix rings. Recently, K. C. O'Meara described an algebra "containing" these examples which makes explaining them a lot easier. Here is the description of that algebra. You can find links to the Bergman examples at that link as well.

Personally, I'm curious about infinite upper-triangular matrix rings. I asked a question about a ring like that but haven't gotten any feedback on it.

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