handle unknown number of variables in Matlab or Mathematica

Question

I have an algorithm to produce an n by n matrix A for any given n. My goal is to find all solutions to the homogeneous system of linear equations Ax=0 using Matlab or Mathematica. I'm trying Matlab first.

Attempt 1: I can't use "linsolve" as A is likely to be singular. I want all the solutions.

Attempt 2: Use solve(eqns, vars, 'ReturnConditions', true). However, how do I turn Ax into equation form? It seems that I need a for loop to put equations into the "eqns", but how can I define symbolic variables x1, x2, ..., xn? I don't know the value of n at first. Here is my psudo-code:

read (n);    
for i=1:n %% i_th equation
       eq=[];
       for j=1:n
         eq=eq+A[i,j]*x[j]; %% keep adding terms
       end
       eq=eq+['==0']; %% add '==0' to make it an equation
       eqns=eqns+eq %% add the ith equation to the equation list
 end
 vars=[];
 for i=1:n 
     vars=vars+x[i];
 end
 solve(eqns, vars, 'ReturnConditions', true)

Can you help making it a real Matlab or Mathematica code?

Brice · Accepted Answer · 2018-12-07 09:20:43Z

1

Matlab is made for linear algebra, so A*x=0 is an equation with one, many or no vector x as a solution.

The problem with Matlab is, it has a great many ways to give you one solution but not to give all solutions. The solution to the linear system Ax=b is usually obtained with x = b\A (where x will have the smallest L0 norm among all solutions, i.e. the fewest non-zero components) or x=pinv(A)*b (where x will have the smallest L2 norm among all solutions). See pinv documentation.

But in this specific case where b=0 , both will give the trivial solution x=0

Now, seeking values of x such that A*x=0 is nothing but looking for the kernel of matrix A. And Matlab null function that does just that.

answered Dec 7, 2018 at 9:20

Brice

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