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

研究启动奖--调查特殊类矩阵和图的组合结构--的目标是找到一种测试，它将确定可约矩阵的最终非负性，并确定其他矩阵类的最终性质。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。由于应用中出现的物理和模型约束，施加或考虑状态的非负性条件是很有意义的。这样的应用与理解A^k随k增加时的行为的问题直接相关。这个项目的目标将改变我们回答关于矩阵的大幂的非负性和可约性的公开问题的方式。矩阵M的一个最终性质是对于所有的幂M^k，k=k0，对于某个正整数k0，幂指数。最终，正矩阵和最终的非负矩阵在控制理论中得到了应用，自1978年引入以来，人们一直在研究它们。对于固定的n，最终正的或最终非负的n×n矩阵的幂指数可能是任意大的，因此不可能通过计算幂来证明一个矩阵不是最终正的或最终非负的。Perron-Frobenius理论展示了检验最终正性的几种方法，2010年，Hogben发现了最终不可约矩阵的最终非负性检验。这项研究的目的是调查剩余的最终非负矩阵类，这些矩阵并不被很好地理解。

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.