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تسجيل مستخدم جديدMinimum ranks of sign patterns via sign vectors and duality
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A {it sign pattern matrix} is a matrix whose entries are from the set ${+,-, 0}$. The minimum rank of a sign pattern matrix $A$ is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of $A$. It is shown in this paper that for any $m times n$ sign pattern $A$ with minimum rank $n-2$, rational realization of the minimum rank is possible. This is done using a new approach involving sign vectors and duality. It is shown that for each integer $ngeq 9$, there exists a nonnegative integer $m$ such that there exists an $ntimes m$ sign pattern matrix with minimum rank $n-3$ for which rational realization is not possible. A characterization of $mtimes n$ sign patterns $A$ with minimum rank $n-1$ is given (which solves an open problem in Brualdi et al. cite{Bru10}), along with a more general description of sign patterns with minimum rank $r$, in terms of sign vectors of certain subspaces. A number of results on the maximum and minimum numbers of sign vectors of $k$-dimensional subspaces of $mathbb R^n$ are obtained. In particular, it is shown that the maximum number of sign vectors of $2$-dimensional subspaces of $mathbb R^n$ is $4n+1$. Several related open problems are stated along the way.
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A emph{sign pattern (matrix)} is a matrix whose entries are from the set ${+, -, 0}$. The emph{minimum rank} (respectively, emph{rational minimum rank}) of a sign pattern matrix $cal A$ is the minimum of the ranks of the real (respectively, rational)
matrices whose entries have signs equal to the corresponding entries of $cal A$. A sign pattern $cal A$ is said to be emph{condensed} if $cal A$ has no zero row or column and no two rows or columns are identical or negatives of each other. In this paper, a new direct connection between condensed $m times n $ sign patterns with minimum rank $r$ and $m$ point--$n$ hyperplane configurations in ${mathbb R}^{r-1}$ is established. In particular, condensed sign patterns with minimum rank 3 are closed related to point--line configurations on the plane. It is proved that for any sign pattern $cal A$ with minimum rank $rgeq 3$, if the number of zero entries on each column of $cal A$ is at most $r-1$, then the rational minimum rank of $cal A$ is also $r$. Furthermore, we construct the smallest known sign pattern whose minimum rank is 3 but whose rational minimum rank is greater than 3.
A sign pattern matrix is a matrix whose entries are from the set ${+,-,0}$. If $A$ is an $mtimes n$ sign pattern matrix, the qualitative class of $A$, denoted $Q(A)$, is the set of all real $mtimes n$ matrices $B=[b_{i,j}]$ with $b_{i,j}$ positive (r
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