Solve a system of linear equations Ax=kb

In summary, the conversation discusses solving a system of linear equations with a matrix A, vector b, and unknown vector x. The matrix A has m*n elements and its rows must sum to 1, with elements between 0 and 1. Similarly, the vector b has m*1 elements between 0 and 1, and the unknown vector x has n*1 elements between 0 and 1. The goal is to find a solution for x that satisfies the equations, with an arbitrary constant k. The conversation also mentions the possibility of using the pseudo-inverse matrix of A, but it may not be a proper solution due to some elements of x being less than 0. The conversation ends with a request for suggestions or
  • #1
turbulent1
3
0
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!
 

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  • #2
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.
 
  • #3
Thank you professor HallsofIvy for your reply!
The attachment is a pdf file, which contains the data in the equations.
 

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  • #4
Could anyone give any hint?
 

What is a system of linear equations?

A system of linear equations is a set of two or more equations that involve the same variables. Each equation represents a line on a graph, and the solution to the system is the point where all the lines intersect.

What is A, x, k, and b in the equation Ax=kb?

A is a matrix representing the coefficients of the variables, x is a column vector representing the variables, k is a scalar constant, and b is a column vector representing the constants on the right side of the equations.

How do you solve a system of linear equations using matrix operations?

To solve a system of linear equations using matrix operations, you need to first represent the system in the form of Ax=kb. Then, you can use matrix operations such as row operations, Gaussian elimination, or inverse matrices to solve for the variables in the column vector x.

What are the different methods for solving a system of linear equations?

Some common methods for solving a system of linear equations include substitution, elimination, and matrix operations. Each method has its own advantages and may be more suitable for certain types of systems.

How do you check the solution to a system of linear equations?

To check the solution to a system of linear equations, you can substitute the values of the variables into each equation and see if they satisfy the equations. Another way is to graph the equations and see if the point of intersection is the same as the solution you found.

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