矩阵盲联合块对角化的代数方法研究与应用

Exact/Approximate joint block diagonalization (EJBD/AJBD) problem is to find a congruence transformation matrix such that the prescribed matrices under such transformation are all exactly/approximately block diagonal matrices with the same nonzero pattern. Exact/Approximate general JBD (EGJBD/AGJBD) problem is to find a block diagonal structure with the maximum number of diagonal blocks such that there exist solutions to the corresponding EJBD/AJBD problem. JBD/GJBD problem can be deemed as a decomposition of a third order tensor, it arises in blind source separation with many important applications. Currently, research on GJBD problem mainly focus on numerical methods, and most of them are based on optimization. In this project, we will study GJBD problem via two algebraic approaches: one is based on certain kinds of matrix commuting; the other is based on the spectral decomposition of matrix polynomial. Theoretically, we shall characterize the existence of nontrial solutions to the EGJBD problem, and give the necessary and sufficient conditions for the uniqueness of the solutions, and analyze the perturbation properties of the EGJBD problem. Algorithmically, we shall develop two classes of efficient, reliable and robust algorithms. In the end, we shall deliver a software package for solving AGJBD problem. In the future, the package may be expanded, maintained, or adopted by some other software.

矩阵精确/近似联合块对角化(EJBD/AJBD)问题是求一个合同变换矩阵使得给定的多个矩阵在这个合同变换下同时精确/近似成为具有相同结构的块对角矩阵。矩阵精确/近似盲联合块对角化(EGJBD/AGJBD)问题是求一个块对角数目最多的块对角矩阵结构使得与之对应的EJBD/AJBD问题有解。JBD/GJBD可以被看为三阶张量的一种分解，其在盲源信号分离中有重要应用。现有的关于GJBD问题的研究主要集中在算法上，且大都是优化类算法。在本项目中，我们将通过两类代数方法研究GJBD问题：一种是基于矩阵交换的；一种是基于矩阵多项式谱分解的。理论上，我们将利用上述两类代数方法刻画EGJBD问题非平凡解的存在条件，给出解唯一的充要条件，并对EGJBD问题进行扰动分析。算法上，发展两类用于求解AGJBD问题的算法，并使其高效、可靠、稳定。最终，形成一个求解AGJBD问题的软件包，以供后续扩展、推广。

矩阵精确/近似联合块对角化(EJBD/AJBD)问题是求一个合同变换矩阵使得给定的多个矩阵在这个合同变换下同时精确/近似成为具有相同结构的块对角矩阵。矩阵精确/近似盲联合块对角化(EBJBD/ABJBD)问题是求一个块对角数目最多的块对角矩阵结构使得与之对应的EJBD/AJBD 问题有解。JBD/BJBD可以被看为三阶张量的一种分解，其在盲源信号分离、独立子空间分析中有重要应用。现有的关于JBD问题的研究主要集中在算法上，且大都是优化类算法。在本项目中，我们研究了两类代数方法：一种是基于矩阵交换的;一种是基于矩阵多项式谱分解的，以及基于矩阵分解的迭代方法。理论上，我们刻画了EBJBD问题非平凡解存在的充要条件，给出解唯一的充要条件，并对EJBD问题进行扰动分析。算法上，发展了若干算法，算法高效、 可靠、稳定。