Are these vectors in the non-negative orthant of some $R^K$? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T09:55:42Z http://mathoverflow.net/feeds/question/68905 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/68905/are-these-vectors-in-the-non-negative-orthant-of-some-rk Are these vectors in the non-negative orthant of some $R^K$? Pawan Aurora 2011-06-27T06:27:40Z 2011-07-20T08:00:19Z <p>Given three sets of vectors $S_i=\left(V_{i1},V_{i2},\ldots,V_{in}\right),i=1,2,3$, s.t. the vectors within a set are pair-wise orthogonal and $V_{ij}\cdot V_{kl}\geq 0$ $\forall i,j,k,l$. Also, $\sum_jV_{ij}=w$ $\forall i$ where $w$ is some unit vector and $V_{ij}\cdot w=V_{ij}\cdot V_{ij}$ $\forall i,j$. It is given that the vectors in $S_i,S_j$ $\forall i\neq j$ lie in $R_{\geq 0}^k$ for some $k$. Can all the three sets of vectors be accommodated in $R_{\geq 0}^K$ for some $K$?</p>