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Given A $\in M_{nxn}(\mathbb{R})$ an anti symmetric matrix $(A^T = -A)$, lets write its columns as : $C_1,...,C_n$, we want to show that there exists n non-negative numbers: $a_1 ,..., a_n$ (which are not all 0) such that $a_1C_1+...+a_nC_n$ is a vector with non-negative coordinates.

I first thought to define the set P as all vectors with non-negative coordinates (over real numbers) and if we can find a vector $v \in P$ with $Av$ is in P then we are done, otherwise I wanted to look over the set $M_{nxn} (\mathbb{R}) $\P and infer the existence over there- I'm not sure how to continue or if the beginning of my solution is valid

I'll accept any solution there is, thanks in advance

Ri-Li
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ned grekerzberg
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1 Answers1

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Your question is very interesting. Skew symmetry is preserved under orthogonal similarity, but being entrywise nonnegative is a basis dependent property. I have yet to understand how to reconcile these two seemingly conflicting properties.

Anyway, you are essentially asking that when $A$ is real skew symmetric, whether there exists a nonnegative and nonzero vector $x$ such that $Ax\ge0$. I'm not sure if there is a simpler and more direct proof, but a stronger result by Tucker (1956) states that there actually exists some $x\ge0$ such that $Ax\ge0$ and $Ax+x>0$.

See, e.g. Tucker's theorem from Farkas lemma on this site, or lemma 3 (Tucker existence lemma) on p.15 of Giorgio Giorgi (2014), Again on the Farkas Theorem and the Tucker Key Theorem Proved Easily.

user1551
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