组合矩阵论中的秩问题

Rank problems on partial matrices, sign patterns and graphs come up in a great number of applications, which are interesting hot and difficult topics involving different fields such as combinatorial matrix theory, numerical analysis and optimization theory. This project will focus on the minimum rank problem of partial matrices, which is an interesting open problem. We will study the relations between the minimum rank of a partial matrix and the number of free entries, positions of free entries, values of constant entries. Minimum ranks of partial matrices with special structures will be studied. Meanwhile, we will study rank problems on sign patterns and graphs such as maximum ranks and minimum ranks of positive-negative-zero patterns, zero-nonzero patterns, zero-nonzero-arbitrary patterns, positive-negative-zero-arbitrary patterns and graphs. We will research problems in this project by using special combinatorial structures and properties of partial matrices, sign patterns and graphs. Research methods and techniques will come from different fields such as combinatorial matrix theory, graph theory, numerical analysis and optimization theory. This project is not only significant in theory but also valuable in many fields such as image processing, data mining and neural networks.

部分矩阵的秩、符号模式的秩和图的秩这三类秩问题源于各类实际应用，是涉及组合矩阵论、数值分析和优化理论等领域的一些有趣的热点、难点问题。本项目将围绕部分矩阵最小秩这一公开问题展开。我们将研究部分矩阵中自由变量的个数、位置、已知元素的值与最小秩的关系以及具有特殊结构的部分矩阵的最小秩等问题。同时，我们将研究符号模式的秩问题和图的秩问题，包括正-负-零模式、零-非零模式、零-非零-任意模式、正-负-零-任意模式等符号模式的最大秩和最小秩、图的最大秩和最小秩。我们将充分利用部分矩阵、符号模式和图的特殊组合结构和性质，结合组合矩阵论、图论、数值分析、优化理论等领域的思想方法和研究技巧来研究本项目中的问题。本项目的研究不仅有重要的理论意义，而且在图像处理、数据挖掘和神经网络等领域也有重要的应用价值。

本项目中，我们刻画了将秩一张量乘积矩阵映射到秩一张量的线性映射的结构; 弱化了一些线性保持问题的经典结果中的条件；提出了元素模式矩阵的概念并刻画了非对称正规元素模式中非零元的最大个数；刻画了给定团数的最大有向图。