双曲矩阵分解的理论、计算、扰动分析及应用研究

This project will study the theory, computations, perturbation analysis, and applications of hyperbolic matrix factorizations. Firstly, we will propose some new hyperbolic matrix factorizations according to the related theory of matrices in the classical and indefinite inner product spaces, the properties of (J_1,J_2)-orthogonal and (J_1,J_2)-unitary matrices, the related results of the existing hyperbolic matrix factorizations, and the characteristics of the related classical matrix factorizations, and then discuss their properties and existence and uniqueness conditions. Secondly, we will explore the computation methods of the new proposed hyperbolic matrix factorizations using the hyperbolic Householder transformation, the hyperbolic Givens rotation, Newton iteration, Jacobi iteration, and their variants, and then construct efficient algorithms. Thirdly, for the norm-wise and component-wise perturbations, we will consider the perturbation analysis of the existing and new proposed hyperbolic matrix factorizations using the classical matrix equation approach, the refined matrix equation approach, the matrix vector equation approach, and their combinations, and then derive the first order and rigorous absolute perturbation bounds, relative perturbation bounds, multiplicative perturbation bounds, and condition number estimators which can accurately reflect the influence of perturbation. Finally, we will discuss the applications of some new hyperbolic matrix factorizations in some problems such as the indefinite least squares problem, the indefinite least squares problem with constraints, the indefinite quadratic optimization problem with constraints, and the generalized indefinite linear model problem.

本项目拟研究双曲矩阵分解的理论、计算、扰动分析及应用等。首先，根据经典内积与不定内积空间中矩阵的相关理论、(J_1,J_2)-正交阵和(J_1,J_2)-酉阵的性质、现有双曲矩阵分解的相关结论以及相关经典矩阵分解的特点等构建新型的双曲矩阵分解，并探讨其性质、存在性与唯一性条件等；其次，借助双曲Householder变换、双曲Givens旋转、牛顿迭代、Jacobi迭代以及它们的变型等探索新构建的双曲矩阵分解的计算方法，构造高效的算法；再次，针对范数型和分量型扰动，利用经典的矩阵方程方法、精致的矩阵方程方法、矩阵向量方程方法以及它们的组合等研究现有与新构建的双曲矩阵分解的扰动问题，以期获得能真切反映扰动影响的一阶与严格的绝对扰动界、相对扰动界、乘法扰动界及条件数估计等；最后，探讨部分双曲矩阵分解在不定最小二乘问题、约束不定最小二乘问题、约束不定二次规划问题、广义不定线性模型问题等上的应用。

双曲矩阵分解是经典内积空间中矩阵分解的拓展，不但本身具有重要的理论研究价值，而且还可以应用于数学领域的一些课题如代数Riccati方程、不定最小二乘问题、最优化问题等，同时，在信息论、物理学、电机工程学等领域也应用广泛。. 本项目主要研究了双曲矩阵分解的理论、计算、扰动分析及应用等。具体研究内容及主要结果如下：(1)提出了新型的广义双曲QR分解并探讨了其扰动分析，扩展了广义QR分解的对应结论，相关结果发表在SCI期刊《Linear Algebra Appl.》之上；(2)系统研究了辛QR分解的扰动分析，扩展了经典QR分解的相应结论，相关结果发表在SCI期刊《Linear Multilinear Algebra》之上；(3)研究了不定最小二问题的分量型与混合型条件数，获得了它们的显式表达式以及易计算的上界，相关结果发表在SCI期刊《Linear Algebra Appl.》之上；(4) 研究了矩阵的广义Cholesky分解与Cholesky-like分解的乘法扰动界与严格扰动界，相关结果发表在SCI期刊《Linear Algebra Appl.》、《Appl. Math. Comput.》之上；(5) 提出了新方法获得了矩阵QR分解、LU 分解、双曲QR分解等新的严格扰动界，相关结果发表或接受发表在SCI期刊《Numer. Linear Algebra Appl.》与《Linear Multilinear Algebra》之上；(6) 研究了非线性方程的条件数及其统计估计、矩阵极分解因子的扰动界、四分块矩阵Drazin逆的表达式、矩阵的Hermitian与半正定广义逆、不定内积空间下的矩阵的Moore-Penorose逆的分量型与混合型条件数，相关结果发表在SCI期刊《Linear Algebra Appl.》、《Math. Inequal. Appl.》、《Indian J. Pure Appl.Math.》、《Filomat》、《J. Comput. Anal. Appl.》之上。此外，研究了矩阵SR分解的乘法扰动界、双曲矩阵分解的修正问题、半定内积下的矩阵奇异值分解问题等。所得结果丰富了双曲矩阵分解及相关问题的理论研究与应用范围，部分结果已被学者引用与推广。