How to move larger values close to matrix diagonal in a correlation matrix

Question

I have a correlation matrix X of five elements(C1,C2,C3,C4,C5)

      C1    C2    C3     C4   C5  

 C1    *     1     0     1     0
 C2    1     *     0              


        

Answer 1

What you're looking for is probably the reverse Cuthill-McKee algorithm (RCM), which pretty much does what you want: for a given matrix it finds a permutation that tends to have its non-zero elements closer to the diagonal. There's a built-in function symrcm in MATLAB that does just that.

So assuming that X is your matrix, you can do the following:

p = symrcm(X);
Xnew = X(p, p);

Xnew is the new reordered matrix, and p is the new row/column order.

Example

Let's create a matrix first:

X = [10 0 0 7 0; 3 20 0 0 11; 0 0 30 0 29; 12 7 0 40 0; 0 33 0 0 50]

Now let's reorder it:

p = symrcm(X);
Xnew = X(p, p)

The result is:

Xnew =    
    40    12     7     0     0
     7    10     0     0     0
     0     3    20    11     0
     0     0    33    50     0
     0     0     0    29    30

Seems right.

Answer 2

A = [1 0  0 1 0;
     0 1  0 0 1;
     0 0  1 0 1;
     1 1  0 1 0;
     0 1  0 0 1]; 
N = length(A);
switched = false;

%%
% Calculate initial Global Energy
disp(A);
global_energy = 0;
for l = 1:N
    for m = 1:N
        if(A(l,m))
            global_energy = global_energy + (l-m)^2/2;
        end
    end
end
disp(global_energy); 

counter = 0;
counter_cutoff = 10000000000;
while(true)
    switched = false;
    counter = counter + 1;
    for i = 1:N
        for j = i+1:N        
            current_metric = 0; % Calculate metric of row i and j with columns i and j
            permuted_metric = 0; % Calculate metric if they were permuted        
            % Row i
            for k = 1:N
                if(k ~= i && k ~= j && A(i,k))
                    current_metric = current_metric + (i-k)^2/2;
                    permuted_metric = permuted_metric + (j-k)^2/2;
                end
            end
            % Row j
            for k = 1:N
                if(k ~= i && k ~= j && A(j,k))
                    current_metric = current_metric + (j-k)^2/2;
                    permuted_metric = permuted_metric + (i-k)^2/2;
                end
            end
            % Col i
            for k = 1:N
                if(k ~= i && k ~= j && A(k,i))
                    current_metric = current_metric + (i-k)^2/2;
                    permuted_metric = permuted_metric + (j-k)^2/2;
                end
            end
            % Col j 
            for k = 1:N
                if(k ~= i && k ~= j && A(k,j))
                    current_metric = current_metric + (j-k)^2/2;
                    permuted_metric = permuted_metric + (i-k)^2/2;
                end
            end

            % If permuted metric is less, swap columns and rows - set switched to true 
            if(permuted_metric < current_metric)
                switched = true; % there was at least one switch
                % Now switch rows and columns
                % Switch columns first
                A(:,[i j]) = A(:,[j i]);
                % Now switch rows
                A([i j],:) = A([j i],:);
            end
        end
    end
    if(~switched || counter > counter_cutoff)
        % All permutations did not lead to a switching of rows and columns
        break;
    end
end

% Calculate final Global Energy
disp(A);
global_energy = 0;
for l = 1:N
    for m = 1:N
        if(A(l,m))
            global_energy = global_energy + (l-m)^2/2;
        end
    end
end
disp(global_energy); 

Terminal:

 1     0     0     1     0
 0     1     0     0     1
 0     0     1     0     1
 1     1     0     1     0
 0     1     0     0     1

22

 1     1     0     0     0
 1     1     1     0     0
 0     0     1     1     0
 0     0     1     1     0
 0     0     0     1     1

 3