generating many random matrices in matlab - matlab

Question

I want to generate 100 different 5 by 5 random matrices using matlab in [0,1] with the following property: Assume that the required matrix is A=[a_{ij}] having the condition a_{ih}+a_{hj}-a_{ij}-0.5=0 (Matrix A is the so-called Fuzzy Preference matrix (i.e.a_{ji}=1-a_{ij} for all i,j), and also consistent). But, I am stuck to write the matlab code. Could anyone help me? Thanks!
Example of such matrix, but not consistent:
A=[.5 .5 .5 .8155 .5 .3423;...
.5 .5 .6577 .8155 .5 .3423;...
.5 .3423 .5 .88662 .75 .3423;...
.1845 .8145 .1338 .5 .25 .25;...
.5 .5 .25 .75 .5 .25;...
.6577 .6577 .6577 .75 .75 .5]
Example of 3 by 3 consistent Fuzzy Preference matrix:
B=[.5 .2 .5;...
.8 .5 .8;...
.5 .2 .5]

Answer 1

To solve this question we need to find solutions to a system of linear equations. The unknowns are the matrix entries, so for an NxN matrix there will be N^2 unknowns. There are N^3 equations since there is an equation for each combination of i, j, and h. This is an overdetermined system, but is consistent, it's easy to see that the constant matrix with all entries 0.5 is a solution.
The system of equations can be written in matrix form, with matrix M defined as follows:
N = 5; % size of resulting matrix
M = zeros(N^3,N^2); % ar = size of matrix A
t = 0;
for i = 1:N
for j = 1:N
for h = 1:N
t = t+1;
M(t,(i-1)*N+h) = M(t,(i-1)*N+h)+1;
M(t,(h-1)*N+j) = M(t,(h-1)*N+j)+1;
M(t,(i-1)*N+j) = M(t,(i-1)*N+j)-1;
end
end
end
The right hand side is a vector of N^3 0.5's. The rank of the matrix appears to be N^2-N+1 (I think this is because the main diagonal of the result must be filled with 0.5, so there are really only N^2-N unknowns). The rank of the augmented system is the same, so we do have an infinite number of solutions, spanning an N-1 dimensional space.
So we can find the solutions as the sum of the vector of 0.5's plus any element of the null space of the matrix.
a = 0.5*ones(N^2,1) % solution to non-homogenous equation
nullspace = null(M) % columns form a basis for the null space
So we can now generate as many solutions as we like by adding multiples of the basis vectors of the null space to a,
s = a+sum(rand(1,N-1).*nullspace,2) % always a solution
The final problem is to sample uniformly from this null space while requiring the all the entries stay within [0,1].
I think it's hard even to say what uniform sampling over this space is, but at least I can generate random elements by the following:
% first check the size of the largest element
% we can't addd/subtract more than 0.5to the homogeneous solution
for p = 1:size(nullspace,2)
mn(p) = 0.5/max(abs(nullspace(:,p)));
end
c = 0;
mat = {};
while c<5 % generate 5 matrices
% a solution is the homogeneous part 0.5*ones(N) plus
% some element of the nullspace sum(rand(1,size(n,2)).*mn.* nullspace,2)
% reshaped to be a square matrix
newM = 0.5*ones(N)+reshape(sum((2*rand(1,size(nullspace,2))-1).*mn.* nullspace,2),N,N)
% make sure the entries are all in [0,1]
if all(newM(:)>=0) && all(newM(:)<=1)
c = c+1;
mat{c} = newM;
end
end
% mat is a cell array of 5 matrices satisfying the condition
I don't have much idea about the distribution of the resulting matrices.