Research Initiation Award Grant: Investigating the combinatorial structure of special classes of matrices and graphs

研究启动奖题为-调查的组合结构的特殊类别的矩阵和图-有一个目标，找到一个测试，将确定最终的非负性的可约矩阵，并确定最终的性质，其他类别的矩阵。在动力系统中，人们经常对关于状态演化的定性信息感兴趣。由于在应用中出现的物理和建模约束，它是感兴趣的施加或考虑的状态的非负性的条件。这样的应用与理解A^k随k增加的行为的问题直接相关。这个项目的目标将改变我们如何回答关于矩阵的大幂次的非负性和约简性的开放性问题。矩阵M的最终性质是对所有幂次M^k，k = k 0成立的性质，对于某个正整数k 0，幂指数。最终正矩阵和最终非负矩阵在控制理论中有应用，自1978年引入以来一直在研究。对于一个固定的n，一个最终为正或最终非负的n × n矩阵的幂指数可以是任意大的，所以不可能通过计算幂来证明一个矩阵不是最终为正或最终非负。Perron-Frobenius理论展示了几种检验最终正性的方法，2010年，Hogben发现了一种检验最终不可约矩阵的最终非负性的方法。本研究的目的是调查的最终非负矩阵，没有得到很好的理解的剩余类。

The Research Initiation Award entitled - Investigating the combinatorial structure of special classes of matrices and graphs - has the goal to find a test that will determine the eventual nonnegativity of reducible matrices and determine eventual properties of other classes of matrices. In dynamical systems, one is frequently interested in qualitative information regarding state evolution. Due to physical and modeling constraints arising in applications, it is of interest to impose or consider conditions for nonnegativity of the states. Such applications are directly linked to the problem of understanding the behavior of A^k as k increases. The objectives in this project will transform how we answer open questions about the nonnegativity and reducibility of large powers of matrices.An eventual property of a matrix M is a property that holds for all powers M^k, k = k0, for some positive integer k0, the power index. Eventually positive matrices and eventually nonnegative matrices have applications to control theory and have been studied since their introduction in 1978. For a fixed n, the power index of an eventually positive or eventually nonnegative n x n matrix may be arbitrarily large, so it is not possible to show a matrix is not eventually positive or not eventually nonnegative by computing powers. Perron-Frobenius theory shows several ways to test for eventual positivity and in 2010 Hogben found a test for eventual nonnegativity for matrices that are not eventually reducible. The objective of this research is to investigate the remaining class of eventually nonnegative matrices that are not well understood.