Find Eigenvalues/Determinant of Infinite Matrix

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    Infinite Matrices
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Discussion Overview

The discussion centers around the possibility of finding eigenvalues or the determinant of an infinite matrix of size \aleph_0 \times \aleph_0. Participants explore theoretical frameworks and conditions necessary for such calculations, particularly in the context of infinite-dimensional spaces.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant questions whether it is possible to find eigenvalues or the determinant of an infinite matrix, suggesting that some matrices might have a finite determinant or be zero based on their entries.
  • Another participant introduces Hilbert-Space theory and operator algebras as relevant frameworks for understanding infinite-dimensional matrices.
  • A third participant notes the necessity of "regularity conditions" for the convergence of infinite sums involved in the analysis of infinite matrices.
  • A later reply reiterates the importance of Hilbert-Space theory and requests recommendations for sources related to these topics.

Areas of Agreement / Disagreement

Participants express varying degrees of uncertainty regarding the conditions under which eigenvalues or determinants can be calculated for infinite matrices. There is no consensus on a definitive approach or solution.

Contextual Notes

Discussion highlights the need for specific regularity conditions for convergence, which remain unspecified. The implications of these conditions on the analysis of infinite matrices are not fully resolved.

cragar
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If I had an infinite matrix [itex]\aleph_0 \times \aleph_0[/itex] could I find the eigenvalues or the Determinant of this matrix. I think some of these matrices would have a finite Determinant or it could be zero. Because i could add 1/2+1/4+1/8... but I would just need a matrix with the right entries. Just wondering if anyone has done this and how you would go about figuring it out.
 
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Hey cragar.

The topic that deals with this kind of thing is the Hilbert-Space theory that deals with operator algebras in infinite-dimensional spaces.

If you want to look into this look into things like Hilbert-Space Theory, Banach Spaces, and operator algebras like C* algebras as well as functional analysis in the infinite-dimensional spaces.
 
Note that you will need to have some kind of "regularity conditions" on the "infinite matrices" in order that the infinite sums involved will converge.
 
chiro said:
Hey cragar.

The topic that deals with this kind of thing is the Hilbert-Space theory that deals with operator algebras in infinite-dimensional spaces.

If you want to look into this look into things like Hilbert-Space Theory, Banach Spaces, and operator algebras like C* algebras as well as functional analysis in the infinite-dimensional spaces.

Can you recommend any sources related to these topics? Thanks.
 

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