MATLAB: How to sum parts of a matrix of different sizes, without using for loops

Question

I have a relatively large matrix NxN (N~20,000) and a Nx1 vector identifying the indices that must be grouped together. I want to sum together parts of the matrix, which in principle can have a different number of elements and non-adjacent elements. I quickly wrote a double for-loop that works correctly but of course it is inefficient. The profiler identified these loops as one of the bottlenecks in my code. I tried to find a smart vectorization method to solve the problem. I explored the arrayfun, cellfun, bsxfun functions, and looked for solutions to similar problems,… but I haven't found a final solution yet. I'll be grateful for any help! ER

This is the test code with the two for-loops:

 M=rand(10); % test matrix
 idxM=[1 2 2 3 4 4 4 1 4 2]; % each element indicates to which group each row/column of M belongs
 nT=max(idxM);
 sumM=zeros(nT,nT);
 for t1=1:nT
  for t2=1:nT
   sumM(t1,t2)=sum(sum(M(idxM==t1,idxM==t2)));
  end
 end

PS: Long-term reader, first-time poster.

Answer 1

   S=sparse(1:N,idxM,1);
   sumM=S.'*(M*S);