Solve for A matrix in Ax = 0

Hi,

I want to solve for a n x n matrix A when I know the n x 1 vector x and the system of linear equations is Ax = 0. The linear constraints I have on solving it is that each column should sum to 0 and I also know the diagonal elements of the A matrix. The matrix A is also tridiagonal, so the only non-zero elements are the ones on the main diagonal and the diagonals above and below it.

I can solve it by hand but unsure about how to do it using MATLAB. I assumed lsqlin would be able to do it but it seems like it only solves for the x vector.

Any ideas will be greatly appreciated. Thanks in advance.

[Torsten]

You have 2*n equations for 2*(n-1) degrees of freedom. So usually, your problem will not have a solution.

Answer 1

[Matt J]

n=numel(x);

mask=tril(triu(ones(n),-1),+1);  %tridiagonal mask

A=optimvar('A',[n,n]);

prob = eqnproblem;
prob.Equations.eqn1=sum(A,1)==0;     %known column sums 
prob.Equations.eqn2=diag(A)==Adiag;  %known diagonal values
prob.Equations.eqn3=A.*mask==A;      %tridiagonal conition
prob.Equations.eqn4=A*x==0;          %A*x==0

[sol,fval,exitflag] = solve(prob);

[Matt J]

In light of the fact that you have more equations than unknowns, you would probably need to seek a least squares solution. It is not clear which of your equations you want to relax from equality to least-squares agreement. If I assume that A*x=0 is that equation, then the code would get modified to,

n=numel(x);

mask=tril(triu(ones(n),-1),+1);  %tridiagonal mask

lb=-inf(n); 
lb(~mask)=0;
lb(1:n+1:end)=Adiag; %known diagonal

ub=-lb;
ub(1:n+1:end)=Adiag;

A=optimvar('A',[n,n], 'Lower', lb,'Upper',-lb);

prob = optimproblem;
prob.Constraints.eqn1=sum(A,1)==0;     %known column sums 
prob.Objective=sum((A*x).^2);          %A*x==0

[sol,fval,exitflag] = solve(prob);

[Torsten]

prob.Equations.eqn1=sum(A,1)==0;     %known column sums 

instead of

prob.Equations.eqn1=sum(A,1)==1;     %known column sums 

?

[Matt J]

Yep, you're right.

[Mainak Dhar]

Thanks all, for the explanations and the solutions. I'm sure I'm framing the question wrong but I can't figure out what. I believe I'll be able to solve it from here by putting in a little work. Much appreciated.