This paper extends some properties of the generalized complementary basic matrices, in particular, in a combinatorial direction. These include inheritance (such as for Alternating Sign Matrices), spectral, and sign pattern matrix (including sign nonsingularity) properties.

A sign pattern (matrix) is a matrix whose entries are from the set {+, −, 0} and a sign vector is a vector whose entries are from the set {+, −, 0}. A sign pattern or sign vector is full if it does not contain any zero entries. 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. The notions of essential row sign change number and essential column sign change number are introduced for full sign...

The sign pattern of a real matrix $A$, denoted by $\mathrm{sgn}A$, is the $(+,-,0)$-matrix obtained from $A$ by replacing each entry by its sign. Let $𝒬\left(A\right)$ denote the set of all real matrices $B$ such that $\mathrm{sgn}B=\mathrm{sgn}A$. For a square real matrix $A$, the Drazin inverse of $A$ is the unique real matrix $X$ such that $A^{k+1}X=A^k$, $XAX=X$ and $AX=XA$, where $k$ is the Drazin index of $A$. We say that $A$ has signed Drazin inverse if $\mathrm{sgn}{\tilde{A}}^{\mathrm{d}}=\mathrm{sgn}{A}^{\mathrm{d}}$ for any $\tilde{A}\in 𝒬\left(A\right)$, where $A^{\mathrm{d}}$ denotes the Drazin inverse of $A$. In this paper, we give necessary conditions for some block triangular matrices to have signed...

The zero forcing number and the positive zero forcing number of a graph are two graph parameters that arise from two types of graph colourings. The zero forcing number is an upper bound on the minimum number of induced paths in the graph that cover all the vertices of the graph, while the positive zero forcing number is an upper bound on the minimum number of induced trees in the graph needed to cover all the vertices in the graph. We show that for a block-cycle graph the zero forcing number equals...

A sign pattern matrix (or nonnegative sign pattern matrix) is a matrix whose entries are from the set $\{+,-,0\}$ ($\{+,0\}$, respectively). The minimum rank (or rational minimum rank) of a sign pattern matrix $𝒜$ is the minimum of the ranks of the matrices (rational matrices, respectively) whose entries have signs equal to the corresponding entries of $𝒜$. Using a correspondence between sign patterns with minimum rank $r\ge 2$ and point-hyperplane configurations in ${ℝ}^{r-1}$ and Steinitz’s theorem on the rational realizability of...

The inertia of an $n$ by $n$ symmetric sign pattern is called maximal when it is not a proper subset of the inertia of another symmetric sign pattern of order $n$. In this note we classify all the maximal inertias for symmetric sign patterns of order $n$, and identify symmetric sign patterns with maximal inertias by using a rank-one perturbation.

The scrambling index of an $n\times n$ primitive Boolean matrix $A$ is the smallest positive integer $k$ such that $A^k{\left(A^{\mathrm{T}}\right)}^k=J$, where $A^{\mathrm{T}}$ denotes the transpose of $A$ and $J$ denotes the $n\times n$ all ones matrix. For an $m\times n$ Boolean matrix $M$, its Boolean rank $b\left(M\right)$ is the smallest positive integer $b$ such that $M=AB$ for some $m\times b$ Boolean matrix $A$ and $b\times n$ Boolean matrix $B$. In 2009, M. Akelbek, S. Fital, and J. Shen gave an upper bound on the scrambling index of an $n\times n$ primitive matrix $M$ in terms of its Boolean rank $b\left(M\right)$, and they also characterized all primitive...