How to find unique solution of X

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Discussion Overview

The discussion revolves around finding a unique solution for the vector X in the context of two matrix equations, A1X = B1 and A2X = B2, where A1 and A2 are 3x3 matrices, and B1 and B2 are 3x1 column vectors. The problem involves exploring methods such as Lagrange multipliers and understanding the implications of the properties of the matrices involved.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Sudipta presents a problem involving two matrix equations and seeks a unique solution for X, expressing difficulty in finding a method to solve it.
  • One participant notes that the system is over-defined, suggesting that unless A2^{-1}B2 = A1^{-1}B1, there is no solution due to the number of equations exceeding the number of unknowns.
  • Sudipta clarifies the intent to use the first equation as a constraint for the second and inquires about combining them using constrained minimization methods.
  • Another participant emphasizes that if A2 is positive definite, it is nonsingular, which would yield a unique solution for A2X = B2, but cautions that this solution must also satisfy A1X = B1.
  • Sudipta provides specific values for B1 and B2, indicating they are different, and shares the matrix A2.
  • A participant argues that no solutions exist by demonstrating that the equations derived from A1 do not satisfy a necessary condition based on the provided B1 values.
  • Sudipta acknowledges the rank deficiency of A1 and the determinant of A2 being zero, indicating complications in finding a solution.

Areas of Agreement / Disagreement

Participants express differing views on the existence of a solution, with some arguing that no solutions exist based on the provided equations, while others explore the potential for finding a solution under specific conditions. The discussion remains unresolved regarding the feasibility of obtaining a unique solution.

Contextual Notes

Participants note the rank deficiency of A1 and the determinant of A2 being zero, which may limit the ability to find a solution. The implications of these properties on the existence of solutions are not fully resolved.

sudipta4
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Hi All,

I have a simple problem

consider two matrix equations
A1X=B1
A2X=B2

A1 and A2 are 3X3 matrix. Moreover, A2 is a positive, definite, symmetric matrix. B1 and B2 are two 3X1 column vectors. The A1, A2, B1, B2 are known. I need to find unique solution of X vector from these two equations. Is there any suitable way to do that?

Can I do it by using Lagrange multiplier? It is not a homework problem. I am struggling this by two days but didn't get success.

Thanks in advance


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Sudipta
 
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Your system is over-defined, Each of your two matrix equations represents three equations, so you have a total of 6 equations and only three unknowns. Let's assume A1 and A2 are non-singular. Unless:
A2^{-1}B2 = A1^{-1} B1
there is no solution. If this is satisfied, then
X = A2^{-1}B2 = A1^{-1} B1
 
Hi,
Thank you very much for your reply.

Actually, I want to use first equation is a constraint for second one. If so how do I combine the first equation with the second equation by some constrained minimization method like Lagrange multiplier method.

in fact, my two matrices are
A1=[[-1 -1 0] [1 0 -1] [0 1 1]]
A2=[[a11 a12 a13] [a12 a22 a23] [a13 a23 a33]]

can you give some clue, how do I proceed?
 
Do you know B1 and B2? If so, what are they?
 
it A2 is positive definite, it s nonsingular, you have a unique solution of A2X = B2.

If that solution does not satisfy A1X = B1, you need to reconsider what you are really trying to do here. You can't "do something impossible" by trying to use A1X = B1 as constraint equations, or whatever.
 
I know the values of B1 and B2 but they are different
B1=[-12.212 0.56 0.65]

B2=[-68.49 -68.49 0.0]
Also the A2 matrix is
A2=[[151.8 2.4 -149.4][2.4 151.8 149.4] [-149.4 149.4 298.8]]

These values are related with my system
 
Last edited:
It's easy to see that no solutions to your problem exist. Expand your equation A1 X = B1, to get:
-x1 - x2 = b1
x1 - x3 = b2
x2 + x3 = b3

Now add all three equations together, giving 0 = b1 + b2 + b3. Your B1 does not satisfy this equation. Therefore no solutions exist.
 
Yes, phyzguy, you are right. I know that. Moreover, A1 is rank deficient and determinant of A2 is 0. therefore, it will not be giving a solution in a simple way. but i need that solution. please help me out in this regard. please give me some clue if anyone aware about this kind of problem.
 

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