A random symmetric matrix whose row and column sum is equal to one and diagonal is zero

i want a random symmetric matrix whose row and column sum is equal to one and diagonal is zero. (The application of this matrix is in the weight graph )

Answer 1

[John D'Errico]

If it is symmetric, AND the row sums are 1, then so are the column sums.

But if you are not picky about the distribution of the elements of the matrix. then it is trivial to do. I'll assume from your comments that you want a n all-nonnegative matrix.

N = 5; % the dimension of the matrix
X = randn(N,2);
M = (X(:,1) - X(:,1).').^2 + (X(:,2) - X(:,2).').^2;

; % M is symmetric, with a zero diagonal. Now scale the matrix.

% using an iterative scheme...
flag = true;
while flag 
  s = sqrt(1./sum(M,1));
  flag = norm(s - ones(1,N)) > 1e-16;
  M = s.*M.*s.';
end
M

M = 5×5

0 0.2997 0.0976 0.2464 0.3562 0.2997 0 0.2287 0.2415 0.2300 0.0976 0.2287 0 0.3860 0.2877 0.2464 0.2415 0.3860 0 0.1261 0.3562 0.2300 0.2877 0.1261 0

% row sums
sum(M,2)

ans = 5×1

1.0000 1.0000 1.0000 1.0000 1.0000

% column sums
sum(M,1)

ans = 1×5

1.0000 1.0000 1.0000 1.0000 1.0000

Easy enough. I imagine I could come up with a scheme with minor thought that is not iterative, but the above will be convergent, and I don't see a reason to make something computationally elegant when there is no gain from doing so. Is the distribution of the elements of that matrix anything special? They are certainly not uniform. GOD NO! But then, you never specified a distribution.

[mohammad]

Thank you very much sir. Your algorithm worked in MATLAB 2021 but gives an error in MATLAB 2014. Can you fix it?

Error using -

Matrix dimensions must agree in line M = (X(:,1) - X(:,1).').^2 + (X(:,2) - X(:,2).').^2;

because size(X(:,1)) is 5*1 but size size(X(:,1).') is 1*5

[John D'Errico]

Just use bsxfun instead. That line of code requires R2016b or later, when the idea of a singleton dimension expansion was introduced. This should work:

M = bsxfun(@minus,X(:,1),X(:,1).').^2 + bsxfun(@minus,X(:,2),X(:,2).').^2;