Linear Algebra Technique for Identifying Impactful Elements in a System

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SUMMARY

The discussion focuses on identifying impactful elements in a control variable vector U that affect a perturbed state variable vector X, represented by the equation X = A * U, where A is a constant coefficients matrix. The participants emphasize the importance of matrix properties, such as diagonalizability and invertibility, in determining the most effective adjustments to U. Techniques mentioned include using the pseudoinverse for minimal L2 norm adjustments and nonlinear optimization for minimal L1 norm adjustments. Additionally, singular value decomposition (SVD) is suggested as a potential method for further analysis.

PREREQUISITES
  • Understanding of matrix operations, specifically matrix multiplication and properties.
  • Familiarity with concepts of matrix diagonalization and invertibility.
  • Knowledge of pseudoinverse and its application in linear algebra.
  • Basic principles of singular value decomposition (SVD) in linear algebra.
NEXT STEPS
  • Explore the application of pseudoinverse in linear regression and its implications for adjustments in U.
  • Research the process of diagonalization of matrices and its significance in simplifying linear transformations.
  • Learn about nonlinear optimization techniques for achieving minimal L1 norm adjustments.
  • Investigate the use of singular value decomposition (SVD) for analyzing the impact of control variables on state variables.
USEFUL FOR

Mathematicians, data scientists, and engineers interested in linear algebra applications, particularly in systems analysis and optimization of control variables in mathematical models.

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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