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.********

本文的研究主要集中在矩阵理论方面，也包含了一些图论的内容。下面讨论的问题出现在量子信息理论、计算机科学、社会网络分析等领域，并且是独立感兴趣的。*特征值逆问题。这里的目标是描述给定的一组对称矩阵的所有可能的特征值，这些对称矩阵具有固定的零-非零模式。零点-非零点模式可以看作是一个图形。这个问题已经在不同的方向得到了广泛的研究，如特征值的数值、特征值的重数和矩阵的秩等。研究了本征值的多重性及相关问题。对于给定的n个顶点的图，人们可能会问，作为该图的特征值的重数列表，可以实现n的哪些整数划分。对于某些图族，如完全图，答案是已知的。然而，这个问题仍然悬而未决。首先，考虑将n划分为两个整数。于是问题就变成了哪些图可以恰好有两个不同的特征值。我们有了几个结果，并正在努力解决整个问题。另一种方法是研究图的特征值的最大重数。通过使用Schur补方法，我们提供了一个确定树和圈的零向量的最大重数和结构的简单方法。我计划在可能的情况下将这种方法推广到所有的图。*全正矩阵。主要目标是解决完全正完成问题。也就是说，给定一个既有指定项又有未指定项的矩阵，是否可以用值替换未指定项，以使结果矩阵中任何阶子矩阵(次项)的行列式为正。我已经完全解决了一阶和二阶未成年人呈阳性的案件。对于较大的未成年人来说，这个问题仍然悬而未决。由于每个子项都是正的，所以每个未指定的项都受一组多项式的限制，这些多项式涉及矩阵的指定项。因此，给定部分矩阵的完全正完成等价于询问这些多项式不等式是否有非空交，当未指定条目的数量增加时，这是具有挑战性的。结果表明，在一个完全正的矩阵中，无论条目的值是多少，一些子式都比其他的大。我打算试着找到未成年人之间的所有这样的关系。当一阶和二阶的子代是正的时，我还打算寻找与子代对应的置换上的偏序，就像Bruhat序在完成问题中所做的那样。有两个项目。其中之一是确定关于有向图和2-边着色图的哪些结果可以推广到混合图。另一个项目是寻找各种类型的独立控制顶点--临界图的结构性质和构造。