Solving equations with singular matrix

In summary, the conversation discusses a problem with solving an equation where A is a known matrix, y is a known column vector, and x is an unknown column vector. The issue is that A is singular, so the simple solution of inverse(A)*y=x cannot be used. The suggested solution is to use SVD (singular Value Decomposition), which is covered in Strang's book. It is mentioned that if A is singular, there may be either no solution or an infinite number of solutions for Ax=y. The conversation also raises questions about the desired solution in case of infinite solutions or no solution at all.
  • #1
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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  • #2
You could look at SVD (singular Value Decomposition). It's covered very nicely in Strang's book
 
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Likes Zak
  • #3
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
 

1. What is a singular matrix?

A singular matrix is a square matrix that does not have an inverse. This means that it cannot be solved using traditional methods of matrix algebra.

2. Why is solving equations with singular matrices difficult?

Solving equations with singular matrices is difficult because without an inverse, the system of equations cannot be solved using standard methods. This often requires the use of more advanced techniques or a different approach altogether.

3. Can a singular matrix be solved?

Yes, a singular matrix can be solved using specialized methods such as the pseudoinverse or by using numerical methods like Gaussian elimination. However, these methods may not always provide a unique solution.

4. What are some real-life applications of solving equations with singular matrices?

Solving equations with singular matrices is commonly used in fields such as physics, engineering, and economics. For example, it can be used to solve systems of equations in circuit analysis, optimization problems in economics, or to model physical systems with multiple variables.

5. How can I determine if a matrix is singular or not?

A matrix can be determined to be singular by calculating its determinant. If the determinant is equal to zero, the matrix is singular and cannot be solved using traditional methods. Additionally, a matrix can also be considered singular if it has linearly dependent rows or columns.

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