Question

I want to create a matrix A [4×8] as follows.

The matrix A always has 1 as its diagonal. A11,A22,A33,A44 = 1

This matrix can be considered as two halves with first half being the first 4 columns and the second half being the second 4 columns like something below :

        1 -1 -1 -1   1 0 0 1
  A =  -1  1 -1  0   0 1 0 0
       -1 -1  1  0   1 0 0 0 
       -1 -1 -1  1   1 1 0 0

Each row in the first half can have either two or three -1’s:

• if it has two -1‘s then that corresponding row in the second half should have one 1
• if any row has three -1‘s the second half of the matrix should have two 1‘s.

The overall objective is to have the sum of each row to be 0. I need to generate all possible combinations of a matrix like this.

It will be better if the matrix with new combination be created at each iteration so that after using it I can discard it or else storing all the combinations is very space intensive. Can anybody help me ?

one possible solution I could think of is to generate all possible combinations of row1, row2, row3 and row4 and create a matrix in each iteration. Does that look feasible?

Answer 1

Here’s one possible solution. If you ignore the diagonal ones for the moment, you can generate all possible patterns for the other 7 values using the functions KRON, REPMAT, PERMS, UNIQUE, EYE, and ONES:

>> rowPatterns = [kron(eye(3)-1,ones(4,1)) ...      %# For 2 out of 3 as -1
                  repmat(eye(4),3,1); ...           %# For 1 out of 4 as 1
                  repmat([-1 -1 -1],6,1) ...        %# For 3 out of 3 as -1
                  unique(perms([1 1 0 0]),'rows')]  %# For 2 out of 4 as 1

rowPatterns =

     0    -1    -1     1     0     0     0
     0    -1    -1     0     1     0     0
     0    -1    -1     0     0     1     0
     0    -1    -1     0     0     0     1
    -1     0    -1     1     0     0     0
    -1     0    -1     0     1     0     0
    -1     0    -1     0     0     1     0
    -1     0    -1     0     0     0     1
    -1    -1     0     1     0     0     0
    -1    -1     0     0     1     0     0
    -1    -1     0     0     0     1     0
    -1    -1     0     0     0     0     1
    -1    -1    -1     0     0     1     1
    -1    -1    -1     0     1     0     1
    -1    -1    -1     0     1     1     0
    -1    -1    -1     1     0     0     1
    -1    -1    -1     1     0     1     0
    -1    -1    -1     1     1     0     0

Note that this is 18 possible patterns for any given row, so your matrix A can have 18^4 = 104,976 possible row patterns (quite a bit). You can generate every possible 4-wise row pattern index by using the functions NDGRID, CAT, and RESHAPE:

[indexSets{1:4}] = ndgrid(1:18);
indexSets = reshape(cat(5,indexSets{:}),[],4);

And indexSets will be a 104,976-by-4 matrix with each row containing one combination of 4 values between 1 and 18, inclusive, to be used as indices into rowPatterns to generate a unique matrix A. Now you can loop over each set of 4-wise row pattern indices and generate the matrix A using the functions TRIL, TRIU, EYE, and ZEROS:

for iPattern = 1:104976
  A = rowPatterns(indexSets(iPattern,:),:);  %# Get the selected row patterns
  A = [tril(A,-1) zeros(4,1)] + ...          %# Separate the 7-by-4 matrix into
      [zeros(4,1) triu(A)] + ...             %#   lower and upper parts so you
      [eye(4) zeros(4)];                     %#   can insert the diagonal ones
  %# Store A in a variable or perform some computation with it here
end