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Partitioning a (1,-1)square matrix into (0,-1,1)summand matrices

asked 2022-01-06 16:25:20 +0100

sgmth gravatar image

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....

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slelievre gravatar imageslelievre ( 2022-01-06 18:20:52 +0100 )edit

Do you require the matrices $A_j$ to be pairwise distinct?

That way there would be finitely many tuples of matrices $A_j$ summing to M.

One could also forbid having $A_j = 0$ for any $j$, and forbid having $A_j = -A_k$ for any $j$ and $k$.

slelievre gravatar imageslelievre ( 2022-01-06 18:24:32 +0100 )edit

Homework ?

Emmanuel Charpentier gravatar imageEmmanuel Charpentier ( 2022-01-06 19:17:11 +0100 )edit

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answered 2022-01-07 04:31:12 +0100

slelievre gravatar image

For each nonnegative integer $k$, you could find the possible tuples of matrices $(A_1, ..., A_k)$ 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 $n^2$ entries $m_{i, j}$; so the $(A_1, ..., A_k)$ are obtained by combining all the possible ways to get each entry.

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Asked: 2022-01-06 16:25:20 +0100

Seen: 30 times

Last updated: Jan 07