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

研究启动奖的题目是“研究特殊类别矩阵和图的组合结构”，其目标是找到一种测试，以确定可约矩阵的最终非负性，并确定其他类别矩阵的最终性质。在动力系统中，人们经常对状态演化的定性信息感兴趣。由于应用程序中产生的物理和建模约束，对状态的非负性施加或考虑条件是很有趣的。这些应用与理解A^k随k增加时的行为问题直接相关。这个项目的目标将改变我们如何回答关于大幂矩阵的非负性和可约性的开放性问题。矩阵M的一个最终性质是一个对所有幂M^k都成立的性质，k = k0，对于某个正整数k0，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矩阵的幂指数可能是任意大的，因此不可能通过计算能力来表示一个矩阵最终不是正的或最终不是负的。Perron-Frobenius理论展示了几种检验最终正性的方法，2010年Hogben发现了一个检验最终不可约矩阵的最终非负性的方法。本研究的目的是研究尚未被很好理解的剩余类最终非负矩阵。

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.