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When I try to solve circuit optimize problem I meet the following question: How to judge if matrix $A$ can transfer to matrix $B$ through row swap and column swap in GL(n, 2) through row swap and column swap? It means that whether we can determine that exist type-1 matrices $s_1$, $s_2$, ... $s_k$ such that: $$s_1 \cdot s_2 ... s_j \cdot A \cdot s_{j+1} ... s_{k} = B$$ type-1 matrices are row switching elementary matrices.

I thought many ways to solve this problem like: 1. sum up the matrix by rows and by colums then compare the sums. 2. sort the matrix through by rows and columns. But they all don' t work (1 is wrong and 2 is hard to proof).

Faerie
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  • Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be closed. To prevent that, please [edit] the question. This will help you recognize and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – José Carlos Santos Jun 19 '23 at 07:09
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    Two matrices are equivalent iff their ranks are equal, so if there exist elementary matrices that reduce both $A$ and $B$ to the same rank normal form, then there exist elementary matrices to convert $A$ to $B$. – V.S.e.H. Jun 19 '23 at 10:37
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    @V.S.e.H. The problem has some difference compare to dealing with 'similar matrix' or 'equivalent matrix', because it constraint on type-1 elementary matrices. And also $A$ and $B$ are both on-singular matrices since they are in GL(n, 2). – Faerie Jun 19 '23 at 11:33
  • Aha, gotcha, so your question is whether there exists a check if two matrices are equal up-to a permutation. Have a look here https://math.stackexchange.com/questions/692605/how-to-tell-if-two-matrices-are-equal-up-to-a-permutation – V.S.e.H. Jun 19 '23 at 15:18
  • @V.S.e.H. Thank you so much!! It really help. – Faerie Jun 20 '23 at 02:03

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