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Let $A \in \mathfrak{so}(n,\mathbb{Z})$ be an integer-valued skew-symmetric matrix. Is there an equivalent matrix $A' \in \mathfrak{so}(n,\mathbb{Z})$, s.t. the number $m$ of non-zero entries is minimal (so $m$ is an invariant)? If so, is this minimal set of non-zero entries itself unique? How can they be determined?

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  • $\begingroup$ I don't understand what you mean by the question "is this minimal set of non-zero entries itself unique?" $\endgroup$ – Omnomnomnom Aug 22 '16 at 14:24
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    $\begingroup$ Also, what does it mean to you for matrices to be "equivalent"? $\endgroup$ – Omnomnomnom Aug 22 '16 at 14:24
  • $\begingroup$ I wondered, whether there is a unique matrix $A' \sim A$,s.t. the number of non-zero entries is minimal. By equivalent, I mean that there are unimodular matrices $S,T$, s.t. $A'=SAT$.. (actually I'm not sure if I need similarity here) $\endgroup$ – Bipolar Minds Aug 22 '16 at 14:34
  • $\begingroup$ Note: A matrix that is equivalent to a skew-symmetric matrix need not be skew-symmetric itself. The same is true for similarity. $\endgroup$ – Omnomnomnom Aug 22 '16 at 14:38
  • $\begingroup$ It seems obvious to me that such a matrix $A'$ will not be unique: it will probably suffice to take $S$ and $T$ to be permutation matrices (of permutations with even parity). $\endgroup$ – Omnomnomnom Aug 22 '16 at 14:39

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