Inverse Eigenvalue Problem, Totally Positive Matrices

逆特征值问题，全正矩阵

• 批准号: RGPIN-2019-05275
• 负责人: Nasserasr, Shahla
• 金额: $ 1.17万
• 依托单位: Brandon University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675738
• 关键词: Inverse; Eigenvalue; Problem; Totally; Positive

The proposed research is mainly in matrix theory, and includes some graph theory. The problems discussed below arise in areas like quantum information theory, computer science, analysis of social networks, and are of interest independently. ***Inverse Eigenvalue Problem. Here, the objective is to describe all possible eigenvalues of a given set of symmetric matrices with a fixed zero-nonzero pattern. The zero-nonzero pattern can be viewed as a graph. This problem has been extensively studied in various directions such as numerical values of eigenvalues, multiplicities of the eigenvalues, and ranks of matrices. I study the multiplicities of eigenvalues and related problems. For a given graph on n vertices, one may ask which integer partitions of n can be achieved as a multiplicity list of the eigenvalues of the graph. The answer is known for some families of graphs such as complete graphs. However, the question remains open. To start, consider partitions of n into two integers. The question then becomes which graphs can have exactly two distinct eigenvalues. We have several results and are working to solve the whole problem. Another approach is to study the maximum multiplicity of the eigenvalues of graphs. By using the Schur complement method we have provided a simple procedure to determine the maximum multiplicity as well as the structure of the null vectors of trees and cycles. I plan to generalize this method for all graphs, when possible.***Totally Positive Matrices. The main goal is to solve the totally positive completion problem. That is, given a matrix with both specified and unspecified entries, can the unspecified entries be replaced with values so that the determinant of every submatrix of any order (minor) in the resulting matrix is positive. I have completely solved the case when the minors of order one and two are positive. The question remains open for larger minors. Since every minor is positive, each unspecified entry is restricted by a set of polynomials involving the specified entries of the matrix. Thus, a totally positive completion of a given partial matrix is equivalent to asking if these polynomial inequalities have non-empty intersection, which is challenging when the number of unspecified entries increases. It turns out that in a totally positive matrix some minors are greater than others regardless of the values of the entries. I intend to try to find all of such relationships between minors. I also intend to search for partial orders on permutations that correspond to the minors in the same way that the Bruhat order did in the completion problem when minors of order one and two are positive.***Graph homomorphisms and domination. There are two projects. One of them is to determine which results about oriented graphs and 2-edge coloured graphs can be generalized to mixed graphs. The other project is to find structural properties of, and constructions for, various types of independent domination vertex--critical graphs.********

拟议的研究主要在矩阵理论中，其中包括一些图理论。下面讨论的问题出现在量子信息理论，计算机科学，社交网络分析等领域，并且独立感兴趣。 ***逆特征值问题。在这里，目的是描述具有固定零零模式的给定对称矩阵的所有可能特征值。零非序模式可以看作是图形。该问题已在各个方向上进行了广泛的研究，例如特征值的数值，特征值的多重性和矩阵等级。我研究特征值和相关问题的多重性。对于N顶点上的给定图，可以询问可以将哪些n的整数分区作为图形特征值的多样性列表来实现。答案以某些图形家庭（例如完整图形）而闻名。但是，问题仍然开放。首先，将n分区分为两个整数。然后，问题变成哪些图可以具有两个不同的特征值。我们有几个结果，正在努力解决整个问题。另一种方法是研究图形特征值的最大多样性。通过使用Schur补体方法，我们提供了一个简单的过程，以确定树木和周期的无效载体的最大多重性以及结构。我计划在可能的情况下将此方法概括为所有图。***完全正面的矩阵。主要目标是解决完全积极的完成问题。也就是说，给定具有指定和未指定条目的矩阵，是否可以用值替换未指定的条目，以使所得矩阵中任何顺序（次要）的每个subsatrix的决定因素为正。当命令一号和二的未成年人呈正时，我已经完全解决了这种情况。对于大型未成年人来说，这个问题仍然开放。由于每个未成年人都是正的，因此每个未指定的条目都受到涉及矩阵指定条目的一组多项式的限制。因此，给定的部分矩阵的完全积极的完成等同于询问这些多项式不平等是否具有非空交叉点，当未指定的条目数量增加时，这具有挑战性。事实证明，在完全积极的矩阵中，一些未成年人不论条目的价值如何。我打算尝试找到未成年人之间的所有此类关系。 我还打算以与未成年人相对应的排列搜索部分订单，就像Bruhat订单在完成问题中所做的相同的方式时，当订单一号和二的订单是正时。***图形同构和统治。有两个项目。 其中之一是确定有关方向图和2边彩色图的哪些结果可以推广到混合图。 另一个项目是找到各种类型的独立支配顶点的结构属性和构造 - 临界图。********