结构矩阵多项式的理论与算法及其在图像识别中的应用

Theory and algorithms of structural matrix polynomials have been one of the hot research topics in numerical linear algebra. In this project, we mainly study the sign characteristic theory and algorithms for several kinds of structural matrix polynomials：(1) we define the sign characteristic of non-Hermitian structural matrix polynomial, study its essential relation with the structures of coefficient matrices, contribute to the structural Jordan triples of non-Hermitian structural matrix polynomials, and also consider the generalized Krein signature theory, the sign characteristic-preserving linearization and the structure-preserving algorithms for evaluating sign characteristic, etc.; (2) by applying the distribution and sign characteristic of eigenvalues, we derive the sufficient and necessary conditions for the solvability of the inverse eigenvalue problems with sign characteristic constrains of the permutation T-even and T-palindromic structural matrix polynomials, a general solution expression and fast algorithms; (3) for the model updating problems for T-even and T-palindromic systems, we study changes of the sign characteristic and positivity when eigenvalues are moving along the imaginary axis or the unit circle; (4) we propose backward stable structure-preserving algorithms for solving the eigenvalue problem of generalized block symmetric matrix pencil with the leading coefficient matrix singular, and apply the structure-preserving theory and algorithms to the recognition of distinguishable images.

结构矩阵多项式的理论与算法已经成为数值代数方向研究的热点问题之一．本项目拟研究若干类非Hermite结构矩阵多项式的符号特征理论与保结构算法：(1)定义非Hermite结构矩阵多项式的符号特征，分析其与系统结构之间的内在联系，建立和完善非Hermite结构矩阵多项式的结构Jordan三元组理论，进而研究广义Krein符号差理论，保符号特征线性化，符号特征的保结构算法等；(2)利用特征值的分布及符号特征，推导带符号特征约束置换T-偶和T-回文结构矩阵多项式特征值反问题有(正定)解的充要条件，并且给出通解表达式和快速算法；(3)针对T-偶和T-回文系统的模型修正问题，研究由特征值沿着虚数轴或单位圆随机移动所引起的符号和系统正定性的变化；(4)研究一类广义块对称首项奇异矩阵束特征值问题，设计向后稳定的保结构算法，并且将保结构理论与算法应用于高分辨率图像识别问题．

结构矩阵多项式特征值问题是数值代数的经典研究领域之一，关于其理论与算法的研究受到学术界的长期关注，具有重要的理论意义和应用价值. 本项目主要研究带特殊结构的矩阵多项式的符号特征理论、特征值反问题和模型修正问题，提出并开发对应的有效可靠保结构算法，并且将研究结果应用于图像识别等领域. 主要研究结果包括：(1)定义非Hermite结构矩阵多项式的符号特征，建立和完善非Hermite结构矩阵多项式的结构Jordan三元组理论；(2)利用特征值的分布及符号特征，给出了若干结构矩阵多项式问题的可解性理论和保结构算法；(3)研究了广义块对称首项奇异矩阵束特征值问题的保结构算法，将保结构理论与算法应用于高分辨率图像识别与处理问题. 项目组发表高质量学术论文10余篇，出版英文学术专著1部.