Given a square matrix M with entries from {1,-1} how to find all possible sets of matrices A1,A2,A3.....(with entries from {0,-1,1}), such that M=A1+A2+A3....
For each nonnegative integer k, you could find the possible tuples of matrices (A1,...,Ak) as follows:
for each of -1, 0, 1, find the ways to write it
as a sum of k summands each in {-1, 0, 1}.
the target matrix M has n2 entries mi,j; so the (A1,...,Ak) are obtained by combining all the possible ways to get each entry.
First of all apologies for replying very late as I was busy with some other things and I logged again yesterday only after so many months. I am really sorry for that. Yes, your idea of finding ways to write -1,0,1 as k summands of {-1,0,1} appears to be very good. i will try to make the code accordingly. Thanks a lot.
Asked: 3 years ago
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Last updated: Jan 07 '22
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Do you require the matrices Aj to be pairwise distinct?
That way there would be finitely many tuples of matrices Aj summing to M.
One could also forbid having Aj=0 for any j, and forbid having Aj=−Ak for any j and k.
Homework ?
Question refined at Ask Sage question 63305.