Solving equations with singular matrix

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SUMMARY

The discussion centers on solving the equation Ax=y, where A is a singular matrix. Since A is singular, the standard method of using the inverse is not applicable. Participants suggest utilizing Singular Value Decomposition (SVD) as a potential solution method. It is emphasized that the nature of the solution depends on whether y lies within the column space of A, leading to either no solution or infinitely many solutions.

PREREQUISITES
  • Understanding of linear algebra concepts, particularly matrix operations.
  • Familiarity with Singular Value Decomposition (SVD).
  • Knowledge of column space and its implications in linear equations.
  • Basic understanding of solution sets in linear systems.
NEXT STEPS
  • Study Singular Value Decomposition (SVD) in detail.
  • Explore the implications of column space on solution existence.
  • Learn about methods for selecting specific solutions in cases of infinite solutions.
  • Investigate alternative approaches for solving singular systems, such as regularization techniques.
USEFUL FOR

Mathematicians, data scientists, and engineers dealing with linear systems, particularly those encountering singular matrices in their work.

Zak
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Hi!

I have a problem: I need to solve an equation, Ax=y, where A is a known matrix, y is a known column vector and x is an unknown column vector. Unfortunately, A is singular so I cannot do the simple solution of inverse(A)*y=x. Does anybody know of any way that I can obtain the coefficients for x?

Thanks in advance
 
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You could look at SVD (singular Value Decomposition). It's covered very nicely in Strang's book
 
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If A is singular then Ax=y either has no solution (eg if y is not in the column space of A) or an infinite number of solutions (can add any solution of Ax=0). So you need to figure out what case you have, and what you want to do. If there are infinite solutions, which one do you want? If there is no solution, what do you mean by "solve?"

jason
 

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