Mathematical Sciences: RUI: Extreme Points of Cones of Positive Semi-definite Matrices with Given Sparsity Pattern

本研究涉及矩阵理论中的两个项目。对于第一个投影，G是n个顶点的无向图，M(G)是一个相关的凸锥。主要研究人员将研究极值矩阵的结构，特别是当M(G)具有大秩极值点时。第二个项目涉及确定某些半定矩阵集合的极值点。他们将调查是否仅靠排名和稀疏性模式来确定一个矩阵是否为极值点。矩阵是一个矩形数组，其中可能包含来自各种环的条目。这是一个起源于十九世纪的领域，是代数的一部分。鉴于它的许多应用以及与数学和其他科学技术学科的其他部分的联系，它目前非常活跃。

This research is concerned with two projects in matrix theory. For the first project, G is an undirected graph on n vertices and M(G) is an associated convex cone. The principal investigators will examine the structure of the extreme matrices, particularly when M(G) has extreme points of large rank. The second project is concerned with determining the extreme points of a collection of certain semidefinite matrices. They will investigate whether the rank and sparsity pattern alone determine if a matrix is an extreme point. A matrix is a rectangular array with possible entries from a wide variety of rings. This is a field which originated in the nineteenth century as part of algebra. It is currently very active in view of its many applications and connections with other parts of mathematics and other scientific and technical disciplines.