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`, and `bsxfun` functions, and looked for solutions to similar problems... but I haven't found a final solution yet.

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=size(M,1);
sumM=zeros(max(idxM),max(idxM));
for t1=1:nT
    for t2=1:nT
        sumM(t1,t2) = sum(sum(M(idxM==t1,idxM==t2)));
    end
end

```