A sign pattern $A$ is a $\pm $ sign pattern if $A$ has no zero entries. $A$ allows orthogonality if there exists a real orthogonal matrix $B$ whose sign pattern equals $A$. Some sufficient conditions are given for a sign pattern matrix to allow orthogonality, and a complete characterization is given for $\pm $ sign patterns with $n-1 \le N_-(A) \le n+1$ to allow orthogonality.
An n × n sign pattern A is said to be potentially nilpotent if there exists a nilpotent real matrix B with the same sign pattern as A. Let D_{n,r} be an n × n sign pattern with 2 \geqslant r \geqslant n such that the superdiagonal and the (n, n) entries are positive, the (i, 1) (i = 1,..., r) and (i, i − r + 1) (i = r + 1,..., n) entries are negative, and zeros elsewhere. We prove that for r \geqslant 3 and n \geqslant 4r − 2, the sign pattern D_{n,r} is not potentially nilpotent, and so not spectrally arbitrary., Yanling Shao, Yubin Gao, Wei Gao., and Obsahuje seznam literatury
A matrix $\Cal A$ whose entries come from the set $\{+,-,0\}$ is called a {\it sign pattern matrix}, or {\it sign pattern}. A sign pattern is said to be potentially nilpotent if it has a nilpotent realization. In this paper, the characterization problem for some potentially nilpotent double star sign patterns is discussed. A class of double star sign patterns, denoted by ${\cal DSSP}(m,2)$, is introduced. We determine all potentially nilpotent sign patterns in ${\cal DSSP}(3,2)$ and ${\cal DSSP}(5,2)$, and prove that one sign pattern in ${\cal DSSP}(3,2)$ is potentially stable.
The inertia set of a symmetric sign pattern $A$ is the set $i(A)=\lbrace i(B) \mid B=B^T \in Q(A)\rbrace $, where $i(B)$ denotes the inertia of real symmetric matrix $B$, and $Q(A)$ denotes the sign pattern class of $A$. In this paper, a complete characterization on the inertia set of the nonnegative symmetric sign pattern $A$ in which each diagonal entry is zero and all off-diagonal entries are positive is obtained. Further, we also consider the bound for the numbers of nonzero entries in the nonnegative symmetric sign patterns $A$ with zero diagonal that require unique inertia.
By a sign pattern (matrix) we mean an array whose entries are from the set $\lbrace +,-,0\rbrace $. The sign patterns $A$ for which every real matrix with sign pattern $A$ has the property that its inverse has sign pattern $A^T$ are characterized. Sign patterns $A$ for which some real matrix with sign pattern $A$ has that property are investigated. Some fundamental results as well as constructions concerning such sign pattern matrices are provided. The relation between these sign patterns and the sign patterns of orthogonal matrices is examined.