An n × n ray pattern A is called a spectrally arbitrary ray pattern if the complex matrices in Q(A) give rise to all possible complex polynomials of degree n. In a paper of Mei, Gao, Shao, and Wang (2014) was proved that the minimum number of nonzeros in an n×n irreducible spectrally arbitrary ray pattern is 3n-1. In this paper, we introduce a new family of spectrally arbitrary ray patterns of order n with exactly 3n - 1 nonzeros., Yinzhen Mei, Yubin Gao, Yanling Shao, Peng Wang., and Obsahuje seznam literatury
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 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 ≥ 2 and point-hyperplane configurations in Rr−1 and Steinitz’s theorem on the rational realizability of 3-polytopes, it is shown that for every nonnegative sign pattern of minimum rank at most 4, the minimum rank and the rational minimum rank are equal. But there are nonnegative sign patterns with minimum rank 5 whose rational minimum rank is greater than 5. It is established that every d-polytope determines a nonnegative sign pattern with minimum rank d + 1 that has a (d + 1) × (d + 1) triangular submatrix with all diagonal entries positive. It is also shown that there are at most min{3m, 3n} zero entries in any condensed nonnegative m × n sign pattern of minimum rank 3. Some bounds on the entries of some integer matrices achieving the minimum ranks of nonnegative sign patterns with minimum rank 3 or 4 are established., Wei Fang, Wei Gao, Yubin Gao, Fei Gong, Guangming Jing, Zhongshan Li, Yanling Shao, Lihua Zhang., and Obsahuje seznam literatury