Finding more than one solution for Matrix Multiplication (Ax = b)

10 views (last 30 days)
For some A and b, there are infinite solutions.
Currently, I only know one way to solve for x, which solves for only 1 x. That is, A\b. How do I find more than 1 solution? Is there any I can find specifically 2 solutions?
Thanks, Clark

Accepted Answer

Wayne King
Wayne King on 15 Sep 2012
Edited: Wayne King on 15 Sep 2012
For example:
A = [1 2; 1 2];
b = [3 3]';
x = pinv(A)*b; % one solution
A*x
% the nullspace of A has dimension 1. So just add that vector
% to the solution to get another solution
y = x+null(A);
A*y % another
Yet another
z = 2*null(A);
w = x+z;
A*w
  1 Comment
Clark
Clark on 15 Sep 2012
Ah! Thanks! I was trying to do A\b + null(null(A)) which obviously wasn't working. So, instead, scalar * null(A). Thanks!

Sign in to comment.

More Answers (3)

Wayne King
Wayne King on 15 Sep 2012
Yes, you can use null() to find a vector from the null space

Clark
Clark on 15 Sep 2012
Edited: Clark on 15 Sep 2012
I don't understand how to use null for this purpose. If I use null(A\b) for example, the answer is simply an empty 1x0 matrix. Could you please elaborate? Sorry, I'm a beginner. I don't see what null would have to do with this. I'm trying to figure out how to use it anyhow.
Thank you.
  1 Comment
Wayne King
Wayne King on 15 Sep 2012
No, you just use null() on the matrix. null(A) then you get a basis for the nullspace of A, you can add any linear combination of vectors from the nullspace of A to a particular solution and you get a different solution of the linear system

Sign in to comment.


Clark
Clark on 15 Sep 2012
Okay, so, A\b + null(A) gives me another solution.
But, can I find a third solution? I was expecting there may be some function which accepts an argument to return a certain number of solutions.
Thanks.
  1 Comment
Wayne King
Wayne King on 15 Sep 2012
All the solutions will be a particular solution plus linear combinations of the nullspace. You can easily write a function that gives you a specified number of linear combinations.

Sign in to comment.

Categories

Find more on Linear Algebra in Help Center and File Exchange

Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!