Solve a system of linear equations Ax=kb

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Discussion Overview

The discussion revolves around solving a system of linear equations represented as Ax=kb, where A is a matrix with specific properties, and the goal is to find the unknown vector x. The context includes constraints on the elements of the matrix and vectors involved, as well as the implications of these constraints on the solution.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Post 1 presents the problem of solving the system Ax=kb, detailing the properties of matrix A and vectors b and x, and expresses concern about using the pseudo-inverse due to the constraints on x.
  • Post 2 questions the clarity of the request and notes that the general matrix equation does not guarantee a unique solution, emphasizing that the actual values will significantly influence the solvability.
  • Post 3 acknowledges the response from another participant and provides additional data in a PDF attachment for further clarification.
  • Post 4 seeks hints or guidance on how to approach the problem.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus. There are differing views on the nature of the solution and the implications of the constraints on the matrix and vectors.

Contextual Notes

The discussion highlights the dependence on specific values for the matrix and vectors, as well as the implications of the stochastic nature of matrix A, which may affect the existence and uniqueness of solutions.

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Solve a system of linear equations Ax=kb
A is a matrix with m*n elements,
[tex]A = \left[\stackrel{a_{11}\; \ldots \;a_{1n}}{ \vdots \ \ddots \ \vdots} {a_{m1}\cdots a_{mn} \\} \right][/tex]
[tex]\sum _{j=1} ^{n}a_{ij}=1 ,0\leq a_{ij}\leq1, 1\leq i\leq m,1\leq j \leq n , m > n[/tex]
b is a vector with m*1 elements,
[tex]0 \leq b_{i} \leq 1 \;,\; 1 \leq i \leq m[/tex],
x is the unknown vector with n*1 elements,
[tex]0 \leq x_{j} \leq 1\:,\:1 \leq j \leq n[/tex],
k is an arbitrary constant which makes x satisfy the system of equations.
find the unknown vector x.

I think it's not proper to solve the system by finding the pseudo-inverse matrix of A,
because some elements of x are than 0.

Your suggestions are welcome, thanks!
 

Attachments

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What exactly do you want? You have a general matrix equation, restricted only by the requirement that the sum of each row be 1 (a stochastic matrix?). There is no one solution. How you would solve it, even whether it has a solution, depends strongly on the actual values.
 
Thank you professor HallsofIvy for your reply!
The attachment is a pdf file, which contains the data in the equations.
 

Attachments

Could anyone give any hint?
 

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