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

You can use accumarray as follows:

nT = size(M,1); % or nT = max(idxM)
ind = bsxfun(@plus, idxM(:), (idxM(:).'-1)*nT); % create linear indices for grouping
sumM = accumarray(ind(:), M(:), [nT^2 1]); % compute sum of each group
sumM = reshape(sumM, [nT nT]); % reshape obtain the final result

A solution using cumsum and diff.

[s,is] = sort(idxM);
sumM  = M(is,is);
idx = [diff(s)~=0 ,true];
CS = cumsum(sumM);
CS = cumsum(CS(idx,:),2);
n=sum(idx);
result = diff([zeros(n,1) diff([zeros(1,n); CS(:,idx)])],1,2);
sumM (:)=0;
sumM (s(idx),s(idx))=result;