3D matrix with equation

I have 2 matrices, Y and P, both of size m by n. I need to implement the following equation:

I'm having an issue with the index changing between j and l.

My initial try was:

eq = 2*Y.*Y - Y.*P - Y.*P

However, this is wrong of course since the multiplication would be with the same indices. I was thinking I would have to expand the matrices into a third dimension to overcome this problem with an extra index and then run some similar code for the equation, but I don't really understand how to perform this expansion and how the code for the equation would look like afterwards.

If you know another solution than to use a 3D matrix, this would of course also be much appreciated.

If it helps, I need to sum over index i afterwards, so this will reduce the dimensions again afterwards.

To clarrify: i runs from 1 to m and j & l run from 1 to n.

Answer 1

[Matt J]

If it helps, I need to sum over index i afterwards, so this will reduce the dimensions again afterwards.

Don't multiply-then-sum. Do them in one combined step using matrix multiplications (and transpositions, where appropriate).

eq = 2*Y.'*Y - P.'*Y - Y.'*P     %entries are eq(j,l)

[Wilko]

Thanks, this solves my issue. I did this earlier in a similar situation, but difference in notation together with some reading about 3D matrices earlier messed up my train of thought.

Answer 2

[J Chen]

does mean ? You can calculate the result using three loops on i, j, and l.

Answer 3

[David Hill]

Should be something like this if I understood you correctly.

[m,n]=size(Y);
N=zeros(n);
for j=1:n
   for l=1:n
      N(j,l)=sum(2*Y(:,j)*Y(:,l)'-Y(:,l)*P(:,j)'-Y(:,j)*P(:,l)','all');
   end
end