[Paper Review] Minimum ranks of sign patterns via sign vectors and duality

Marina Arav, Frank J. Hall+4 more|arXiv (Cornell University)|Dec 20, 2013

graph theory and CDMA systems7 references3 citations

TL;DR

This paper introduces a novel approach using sign vectors and duality to analyze the minimum rank of sign pattern matrices. It proves that for any $m \times n$ sign pattern with minimum rank $n-2$, rational realization of the minimum rank is always possible, and further shows that for $n \geq 9$, there exist $n \times m$ sign patterns with minimum rank $n-3$ that cannot be rationally realized. It also characterizes sign patterns with minimum rank $n-1$ and establishes that the maximum number of sign vectors for a 2-dimensional subspace of $\mathbb{R}^n$ is $4n+1$.

ABSTRACT

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 $m imes n$ 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 $n\geq 9$, there exists a nonnegative integer $m$ such that there exists an $n imes m$ sign pattern matrix with minimum rank $n-3$ for which rational realization is not possible. A characterization of $m imes n$ 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.

Motivation & Objective

• Address the longstanding open problem of characterizing sign patterns with minimum rank $n-1$.
• Establish conditions under which the minimum rank of a sign pattern matrix can be rationally realized.
• Extend the understanding of sign vectors of subspaces and their extremal cardinalities.
• Provide a new duality-based framework for analyzing minimum rank problems in sign pattern matrices.
• Solve the open problem of characterizing $m \times n$ sign patterns with minimum rank $n-1$.
• Identify the maximum number of sign vectors for 2-dimensional subspaces of $\mathbb{R}^n$.
• Explore the existence of sign patterns with minimum rank $n-3$ that cannot be rationally realized.

Proposed method

• Use sign vectors of subspaces to characterize the minimum rank of sign pattern matrices.
• Apply duality principles between subspaces and their orthogonal complements to analyze sign vector sets.
• Utilize condensed sign patterns to simplify analysis and preserve minimum rank properties.
• Apply results from combinatorial geometry and projective geometry to bound extremal sign vector counts.
• Construct explicit sign pattern matrices to demonstrate tight bounds on sign vector cardinalities.
• Use reduced row echelon forms to analyze the independence and structure of sign vectors in subspaces.

Experimental results

Research questions

• RQ1 What conditions guarantee rational realization of the minimum rank for sign pattern matrices?
• RQ2 Can the minimum rank of an $m \times n$ sign pattern with rank $n-2$ always be rationally realized?
• RQ3 For which $n \geq 9$ do there exist $n \times m$ sign patterns with minimum rank $n-3$ that cannot be rationally realized?
• RQ4 What is the maximum number of distinct sign vectors realizable by a 2-dimensional subspace of $\mathbb{R}^n$?
• RQ5 How can sign vectors and duality be used to characterize sign patterns with minimum rank $n-1$?
• RQ6 What are the extremal values of the number of sign vectors for $k$-dimensional subspaces of $\mathbb{R}^n$?

Key findings

• For any $m \times n$ sign pattern matrix with minimum rank $n-2$, rational realization of the minimum rank is always possible.
• There exists an $n \times m$ sign pattern matrix with minimum rank $n-3$ that cannot be rationally realized for each $n \geq 9$.
• The maximum number of sign vectors for a 2-dimensional subspace of $\mathbb{R}^n$ is $4n+1$.
• The minimum number of sign vectors for a $k$-dimensional subspace of $\mathbb{R}^n$ is $3^k$, achieved when the subspace is spanned by standard basis vectors.
• A characterization of $m \times n$ sign patterns with minimum rank $n-1$ is provided, solving an open problem from Brualdi et al. [6].
• The maximum number of sign vectors for $(n-1)$-dimensional subspaces of $\mathbb{R}^n$ is $3^n - 2(2^n - 1)$.