Linear Algebra Technique for Identifying Impactful Elements in a System

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Discussion Overview

The discussion revolves around identifying impactful elements in a control variable vector U that affect corresponding state variables in a system represented by a constant matrix A. Participants explore linear algebra techniques for determining which elements in U should be adjusted to correct perturbed elements in X, without relying on algorithmic or programming methods.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant proposes that the maximum element in the corresponding row of matrix A multiplied by U indicates which element in U has the most effect on X.
  • Another participant suggests considering whether matrix A is diagonalizable, as its diagonal form might provide clearer insights.
  • A different viewpoint states that if A is invertible, the adjustment to U is unique, whereas if A is rank deficient, using the pseudoinverse can yield a minimal adjustment in L2 norm, and nonlinear optimization can be used for minimal L1 norm adjustments.
  • One participant inquires about the application of singular value decomposition (SVD) in this context.

Areas of Agreement / Disagreement

Participants express various techniques and considerations, but there is no consensus on a single formal linear algebra technique to address the problem. Multiple competing views and approaches remain present in the discussion.

Contextual Notes

Participants mention conditions such as the invertibility and rank of matrix A, which may affect the proposed techniques. The discussion does not resolve these conditions or their implications.

mhdella
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Let’s say that we have a constant matrix A which is the coefficients matrix and column vector U of control variable as well as column vector X of state variables:
X=A*U
The question is: What is the proper technique in Linear Algebra that I should do to know which element in U has the most impact on the corresponding perturbed element in X.
On other words, there is an element in X has been perturbed and I would like to correct it by adjusting a few (as less as I can) elements in U.
I know the maximum element in the corresponding row of A which is multiplied by U column vector would have the most effect and by that I will know the corresponding element in U, but I am searching about a formal linear algebra technique to deal with this not algorithmic or programming procedure
 
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Just an idea:

Is the matrix A diagonalizable? If so, maybe the diagonal form would make it
clearer .
 
If A is invertible, then the adjustment is unique, if A is rank deficient, the adjustment can be made minimal in L2 norm if you use pseudoinverse, if you want minimal L1 norm adjustment, you go with nonlinear optimization.
 
i appreciate it. Thanx
 
How can I do that by using singular value decomposition?
 

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