Diagonal sums with non-zero elements

Hello,

I have a matrix A, and have computed all possible (n factorial) permutations of this matrix. Next, I want to compute the sum of the diagonals of the permuted matrices containing non-zero elements. I've written a nested for loop to check for non-zero elements, and then, if there are no zero-elements, I want to save the diagonal sum in "diag". But it saves diagonal sums that still contain zero-elements.

All permutations of matrix A, with non-zeros on the diagonal, have constant diagonal sum 1.6. This means that if the code works, it should only find diagonals with sum 1.6

%define A
A = [0 0.4 0.3 0.3; 0 0 0.5 0.5; 0.5 0.3 0 0.2; 0.5 0.3 0.2 0];
[n,n] = size(A);

%find all permutations of A
B = reshape(A(:,flipud(perms(1:4)).'),n,n,factorial(n));

%compute diagonal sums with non-zero elements 
diag = zeros(1, factorial(n));
for i = 1:factorial(n)
    for j=1:n
        if B(j,j,i) == 0
            break
        end 
        diag(1,i) = trace(B(:,:,i));
    end 
end 
diag

Answer 1

[Riccardo Scorretti]

Hi Ingrid,

I guess you mean you want to compute te sum of the diagonals of the permuted matrices containing all non-zero elements. In this case I would program like that:

%define A
A = [0 0.4 0.3 0.3; 0 0 0.5 0.5; 0.5 0.3 0 0.2; 0.5 0.3 0.2 0];

You wrote: [n,n] = size(A). This is weird (for me), I would rather program like that:

[n, m] = size(A);                                   % ***
assert(n == m, 'The matrix A must be square!')      % ***

%find all permutations of A
B = reshape(A(:,flipud(perms(1:4)).'),n,n,factorial(n));

I need to use the built-in function diag, so I had to rename diag to diag_

%compute diagonal sums with non-zero elements
diag_ = zeros(1, factorial(n));                     % diag "shades" a function with the same name
for i = 1:factorial(n)
  if any(diag(B(:,:,i)) == 0) , continue ; end
  diag_(1,i) = trace(B(:,:,i));
end
diag_'

ans = 24×1

0 0 0 0 0 0 0 0 0 1.6000

The error in your program was here:

%define A
A = [0 0.4 0.3 0.3; 0 0 0.5 0.5; 0.5 0.3 0 0.2; 0.5 0.3 0.2 0];
[n,n] = size(A);
%find all permutations of A
B = reshape(A(:,flipud(perms(1:4)).'),n,n,factorial(n));
%compute diagonal sums with non-zero elements 
diag = zeros(1, factorial(n));
for i = 1:factorial(n)
    for j=1:n
        if B(j,j,i) == 0
            

Consider the case: B(:,:,j) = [1 0 0 0 ; 0 2 0 0 ; 0 0 0 0 ; 0 0 0 0]. In the first two iterations B(j,j,i) ~= 0, so you assign to diag(1,i) the trace of B(:,:,i). But at the third iteration, B(3,3,i) = 0: if you break the loop, it is mandatory to reset the value of diag(1,j) to 0 otherwise it will store the wrong result.

By the way, in this algorithm the computation of the trace is uselessly repeated many times.

            diag(1,i) = 0;      % THIS WAS MISSING
            break
        end 
        diag(1,i) = trace(B(:,:,i));
    end 
end 
diag

diag = 1×24

0 0 0 0 0 0 0 0 0 1.6000 1.6000 0 0 0 0 0 1.6000 1.6000 0 0 0 0 1.6000 1.6000

Of course, both versions give exactly the same result.

Answer 2

[Davide Masiello]

%define A
A = [0 0.4 0.3 0.3; 0 0 0.5 0.5; 0.5 0.3 0 0.2; 0.5 0.3 0.2 0];
[n,n] = size(A);

%find all permutations of A
B = reshape(A(:,flipud(perms(1:4)).'),n,n,factorial(n));

%compute diagonal sums with non-zero elements 
dgnl = [];
for i = 1:factorial(n)
    if all(diag(B(:,:,i))~=0)
        dgnl = [dgnl,sum(diag(B(:,:,i)))];
    end 
end 
dgnl

dgnl = 1×6

1.6000 1.6000 1.6000 1.6000 1.6000 1.6000

Better not use diag as variable name, since it's already a MatLab in-built function.