Infinite-dimensional matrix multiplication

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Discussion Overview

The discussion revolves around the properties of infinite-dimensional matrix multiplication, particularly focusing on the conditions under which associativity holds. Participants explore the relationship between infinite matrices and linear maps, as well as the implications of convergence in this context.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants assert that infinite-dimensional matrix multiplication is generally not associative, questioning the criteria under which associativity might be valid.
  • There is a suggestion that the question of associativity may relate to the interchangeability of summation symbols, with a focus on the existence of those sums.
  • One participant mentions the Tonelli and Fubini theorems, indicating that absolute convergence is crucial for interchanging summation in the context of infinite series.
  • Another participant expresses confusion about the associativity of infinite matrix multiplication, suggesting that it should be associative if it corresponds to composing linear maps, but questions the applicability of this in the infinite case.
  • Concerns are raised about the representation of infinite matrices as linear maps, noting that a finiteness condition on non-zero entries is typically required in linear algebra.
  • Clarifications are made regarding the distinction between analysis and linear algebra, emphasizing that convergence issues are primarily relevant in analysis.
  • One participant corrects a previous mention of a theorem, confirming the correct name as Tonelli.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the conditions for associativity in infinite-dimensional matrix multiplication. Multiple competing views and uncertainties remain regarding the relationship between infinite matrices, linear maps, and convergence.

Contextual Notes

Participants highlight the need for precision in defining terms like "associative" and the context of the discussion, indicating that the treatment of infinite matrices may vary significantly between linear algebra and analysis.

alex5
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We know that infinite-dimensional matrix multiplication in general isn't asociative. But, is there any criteria when asociativity is valid?
thanks in advance.
 
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Well, isn't this the same thing as asking when you can interchange two summation symbols?
 
Yes, I think you'r right. Can we say: we can interchange two summation symbols if both exists?
 
Some buzzwords and a book

Relevant buzzwords include Tonelli theorem and Fubini theorem. In terms of doubly indexed infinite series, these basically say that absolute convergence is the key. See Bartle, The Elements of Real Analysis, 2nd Edition, Wiley, 1976, section 36, for an undergraduate level discussion.
 
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I am puzzled. If multiplying infinite matrices corresponds to composing linear maps, as in the finite dimensional case, then it seems it would be associative, since composition is so.

I guess these matrices do not correspond to linear maps in the algebraic sense, as in that case there would be a finiteness condition on the number of non zero entries in the columns.
 
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mathwonk said:
I am puzzled. If multiplying infinite matrices corresponds to composing linear maps, as in the finite dimensional case, then it seems it would be associative, since composition is so.

I guess these matrices do not correspond to linear maps in the algebraic sense, as in that case there would be a finiteness condition on the number of non zero entries in the columns.

Consider, for example:

[tex]a_{ij}=\frac{1}{2^{i+j}}[/tex]
[tex]b_{ij}=1[/tex]
[tex]c_{ij}=\frac{1}{i+j}[/tex]

Some products will be convergent, and some will be divergent.
 
one needs to be a little more precise. in linear algebra, convergence is not an issue, only in analysis.

so one needs to say what subject one is working in, and what one means by "associative".

the matrices you gave do not represent maps in terms of a basis in the linear algebra sense.
 
mathwonk said:
one needs to be a little more precise. in linear algebra, convergence is not an issue, only in analysis.

so one needs to say what subject one is working in, and what one means by "associative".

the matrices you gave do not represent maps in terms of a basis in the linear algebra sense.

I meant that the elements in of the matrix are divergent sums.
 
mathwonk said:
(Is it perhaps Tonelli?)

Yes, indeed, thanks. I have made the correction.
 

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