Computing the Inverse of a Cubic Matrix

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    Cubic Inverse Matrix
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Discussion Overview

The discussion revolves around the computation of the inverse of a cubic matrix, specifically a 2x2x2 tensor. Participants explore the definitions and implications of invertibility in the context of higher-dimensional matrices and tensors, as well as potential applications in solving systems of conic equations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant asks how to compute the inverse of a cubic matrix defined by [aijk].
  • Another participant suggests that a clear definition of the inverse of a cubic matrix is necessary before proceeding.
  • A participant questions whether a cubic matrix represents a 3-linear map and notes uncertainty regarding the concept of invertibility for k-tensors.
  • One participant proposes that the inverse of a cubic matrix could be relevant for solving systems of conic equations, suggesting a potential similarity to the quadratic formula.
  • Another participant recommends attempting an analytical solution and reformulating it in matrix form, while expressing doubt about the applicability due to the non-linear nature of the problem.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the definition or the feasibility of computing the inverse of a cubic matrix, with multiple competing views and uncertainties expressed throughout the discussion.

Contextual Notes

Limitations include the lack of clarity on the definition of the inverse for cubic matrices and the implications of non-linearity in the problem being discussed.

Bruno Tolentino
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Be the matrix A defined by [aijk] (a matrix 2x2x2), do you know how to compute a inverse this cubic matrix?
 
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First you would need to define what you mean by the inverse of a cubic matrix.
 
Is the cubic matrix (the representation of ) a 3-linear map? If so, then it is a 3-tensor. I don't think there is a notion of invertibility for k-tensors, but I am not sure.
 
The inverse of a cubic matrix is for try to solve a system of conic equation, like this:

002.jpg


I think that the solution could have the same form that quadratic formula, so, would be necessary to find the inverse the matrix A...
 
If you don't find formulas, you can try the same approach as for normal matrices: solve analytically and write the final solution in matrix form again.
I'm not sure if that works here as your problem is not linear.
 

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