矩阵张量积空间上的线性保持问题

发布时间：2018-06-16 04:39

本文选题：线性保持问题 + 张量积空间　；参考：《哈尔滨工业大学》2016年博士论文

【摘要】：在理论数学中,不变量的研究占据着重要的地位。保持问题是在一个给定的数学结构上研究保持某种不变量的映射的问题。在矩阵理论中,保持问题被明确地提出并成为矩阵理论中的一个核心研究领域。学者们对多种不变量的线性保持问题进行研究,并且也从多个角度将线性保持问题进行推广。在2012年矩阵与算子国际会议上,时任国际线性代数协会副主席李志光教授结合量子信息科学的背景,提出在矩阵张量积空间上线性保持问题的描述方式,特别地,明确地提出了保矩阵张量积秩(更一般的,保秩1)的公开问题。该类问题将不变量的范围限制到纯张量集合,使映射的约束减少,从而期望得到更宽泛的映射的形式,但同时问题的研究也变得困难。本文围绕矩阵张量积空间上的线性保持问题展开研究。论文研究内容包括以下三个方面:(1)研究保矩阵张量积秩的线性映射。通过例子说明保矩阵张量积秩的线性映射一般不再是保矩阵秩的线性映射。在矩阵张量积空间上定义典范映射。对典范映射的基本性质进行了研究。刻画保矩阵张量积秩的线性映射结构,进而解决李志光教授提出的一个公开问题。(2)研究保Hermite矩阵张量积秩1的线性映射。对Hermite矩阵张量积空间上典范映射的基本性质进行研究。在Hermite矩阵张量积空间中构造纯张量秩1阵的集合升链,得到一类由典范映射所决定的线性映射。刻画保Hermite矩阵张量积秩1的线性单射,并举例说明单射的必要性。(3)研究保(Hermite)矩阵张量积幂等的线性映射。通过构造纯张量幂等Hermite矩阵集合升链,运用映射延拓和限制的方法,刻画保(Hermite)矩阵张量积幂等的线性映射。作为应用,刻画保矩阵张量积立方幂等、M-P逆、群逆的线性映射。本文所研究的内容是经典线性保持问题中的两个核心问题,保秩问题和保幂等问题,在矩阵张量积空间上的推广。这些工作丰富了保持问题在矩阵张量积空间上的现有理论。
[Abstract]:In theoretical mathematics, the study of invariants occupies an important position. The maintenance problem is the problem of preserving the mapping of certain invariants on a given mathematical structure. In matrix theory, the maintenance problem is explicitly proposed and becomes a core research field in matrix theory. In this paper, the linear preserving problem of many invariants is studied, and the linear preserving problem is generalized from many angles. At the 2012 International Conference on Matrix and operators, Professor Li Zhiguang, then Vice President of the International Association of Linear Algebras, proposed a way to describe the problem of linear preservation in matrix tensor product spaces, especially in the context of quantum information science. The open problem of preserving matrix tensor product rank (more generally, rank 1) is presented. This kind of problem limits the scope of invariant to pure Zhang Liang set, reduces the constraint of mapping, and expects to obtain a broader mapping form, but at the same time, the study of the problem becomes difficult. This paper focuses on the linear preserving problem in matrix tensor product space. In this paper, we study the linear mapping preserving the rank of tensor product of matrix in the following three aspects: 1: 1. It is shown by an example that the linear mapping preserving the rank of matrix tensor product is no longer a linear mapping preserving the rank of matrix. The canonical mapping is defined on the matrix tensor product space. The basic properties of canonical mappings are studied. This paper describes the structure of linear mappings preserving the rank of tensor product of a matrix, and then solves an open problem put forward by Professor Li Zhiguang.) We study the linear mappings preserving the rank 1 of the tensor product of Hermite matrices. The basic properties of canonical mappings on tensor product spaces of Hermite matrices are studied. A set ascending chain of pure Zhang Liang rank 1 matrices is constructed in the Hermite matrix tensor product space, and a class of linear mappings determined by canonical mappings are obtained. This paper describes the linear monomorphism preserving the rank 1 of Hermite matrix tensor product, and illustrates the necessity of monomorphism. 3) the linear mapping of tensor product idempotent of preserving Hermite matrix is studied. By constructing the ascending chain of pure tensor idempotent Hermite matrices and using the method of extension and restriction of mappings, this paper describes the linear mapping of tensor product idempotent preserving Hermite matrices. As an application, we characterize the linear mappings of the M-P inverse and group inverse of the tensor product cubic idempotent inverse of the preserving matrix. In this paper, we study two core problems in classical linear preserving problem, rank preserving problem and idempotent preserving problem, which are generalized in matrix tensor product space. These work enrich the existing theory of preserving problem in matrix tensor product space.
【学位授予单位】：哈尔滨工业大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O151.21;O183.2

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