Generate Matrices for Which the Sum is a Sparse Matrix

Hi everyone,

I am trying to generate this matrix:

This matrix is the sum of p+1 matrices. To have a zero in some cell (i,j) of the matrix, there should be a zero in the same cell (i,j) and its transpose _(j,i)_for every Yk matrix. These matrices are calculated like this:

and like this:

The B's are randomly generated lower triangular matrices with entries 0, 0.5, and -0.5.

I would like to know how to constrain the Yk's to have 0's in some cells and their transposes.

I tried the code below but it's not working.

Any suggestion will be very appreciated.

function [Z, omega, tmat] = may195(n,p)
o = pi*2*(rand(1)-0.5);
F = zeros(n, n, p);
A = zeros(p*n, p*n);
check1 = 0;
while check1 == 0;
for s = 1:n;
      for t = 1:n;
          for i = 1:p;
              M = randi(3,n,n);
              M(M==1) = 0.5;
              M(M==2) = 0;
              M(M==3) = -0.5;
              M = -triu(M)';
              F(:,:,i) = M;
          end
          A(1:n,1:p*n) = reshape(F,[n,p*n]);
          A(n+1:p*n, 1:(p-1)*n) = eye((p-1)*n);
          while all(abs(eig(A))) < 1
              U = eye(size(F,2));
              p = size(F,3);
              for l = 1:p;
                  U = U + F(:,:,l)' * F(:,:,l);
              end
              while U(s,t) == U(t,s) && U(s,t) == 0
                  Binit = eye(size(F,2));
                  Y = zeros(size(F,2));
                  Yp = zeros(size(F,2));
                  j = sqrt(-1);
                  for k = 1:p;
                      B = Binit;
                      B = B' * F(:,:,k);
                      for l = 1:(p-k);
                          B = B + F(:,:,l)' * F(:,:,k+l);
                      end
                      Yn = 2*B;
                      Ynn = exp(-j*k*o)*Yn;
                      Y = Y + Ynn;
                      Yt = Yn';
                      Ytt = exp(j*k*o)*Yt;
                      Yp = Yp + Ytt;
                  end
                  while B(s,t) == B(t,s) && B(s,t) == 0;
                      S = U + 0.5 * (Y + Yp);
                      if all(diag(S));
                          check1 = 1;
                      end
                  end
              end
          end
      end
      end
      end
      Z = S;
      omega = o;
      tmat = F;
      end

[Jan]

The description of the problem is rather complicated. All you want to obtain is:

 I would like to know how to constrain the Yk's to have 0's in some cells and their transposes.

Neither the formulas nor the code concern this topic exactly. It is more efficient, if you post the relevant part of the code only and explain, what it does in opposite to your expectations. Currently it seems like you could insert some zeros randomly in your Y matrices.

[Matt J]

I would like to know how to constrain the Yk's to have 0's in some cells and their transposes.

You need to explain what it is that you are allowed to control in imposing these constraints. If, for example, we take the simplified case p=0 where you have only the symmetric matrix Y0, you will make Y0(i,j) = 0 if and only if you can choose B0 so that B0(:,i) is orthogonal to B0(:,j).

[Said Maanan]

@Jan Simon. I think the most relevant parts to my question are these:

     for l = 1:p;
         U = U + F(:,:,l)' * F(:,:,l);
     end

and:

     for l = 1:(p-k);
         B = B + F(:,:,l)' * F(:,:,k+l);
     end

I would like the resulting matrices (Y0 (U) and Yk (B)) to have at least one zero at the same cell (i,j) and its transpose (j,i) before going any further in the computation.