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What is the support of a signed vector? By signed vector, I mean a vector which is determined by considering the signs of the coefficients of the entries of another vector.

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According to TenaliRaman, "support of a vector is the number of non-zero elements in that vector."

I posted it here for clarity.

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Let $\mathbf{v}$ be an $n$-dimensional sign vector, i.e., a vector in $\{+,-,0\}^n$. The the following is a natural partition of $[n]$ induced by $\mathbf{v}$.

  • $\mathbf{v}^+ = \{i \in [n] | v_i = +\}$
  • $\mathbf{v}^- = \{i \in [n] | v_i = -\}$
  • $\mathbf{v}^0 = \{i \in [n] | v_i = 0\}$

The support of $\mathbf{v}$ is the set $[n] \setminus \mathbf{v}^0 = \mathbf{v}^+ \cup \mathbf{v}^-$ consisting of all indices corresponding to nonzero entries in $\mathbf{v}$.

See page 8 of Oriented Matroids for the above definition and the wikipedia page for more generality.

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