对称

Firstly,the generic conception of symmetry for a real sequence was proposed,and then,the symmetry-decomposition and the symmetric degree sequence are presented,which are deduced from the projection theory,the orthogonal-decomposition theory in inner product space and .

首先提出序列信号一般意义下的对称概念，然后由内积空间中的投影、正交分解理论以及内积量化两个信号线性相关程度的特性导出任意信号的对称分解及对称程度序列，对称程度序列定量刻画了信号随对称点的变化时对称特性的变化，在此基础上得出任意序列信号对称程度的定量指标——对称性指标。

Exive least-squares solutions, antire?exive least-squares solutions, bisymmetric least-squaressolutions, symmetric and antipersymmetric least-squares solutions, symmetric or-thogonal symmetric least-squares solutions, symmetric orthogonal antisymmetricleast-squares solutions and their optimal approximation to the linear matrix equa-tion AX = B, and solve them successfully. 2. For Problem II, we can convert it to another problem of finding the least-squares solutions with the least norm of a new consistent matrix equation. Onthe base of the solutions of Problem I we can apply the iterative method to get

本文所构造的迭代法的优点在于先利用法方程变换将求矩阵方程的最小二乘解转化为求一个相容矩阵方程的解的问题，再利用迭代法对于任意给定的初始矩阵进行迭代，均可在有限步内迭代出所求问题的一个解；可将问题II转化为求新方程的极小范数解的问题，同样用迭代法求解，从而系统且全面地解决了问题I、II在约束矩阵类如中心对称、中心反对称、自反矩阵、反自反矩阵、双对称、对称次反对称、对称正交对称、对称正交反对称矩阵中的最小二乘解及其最佳逼近问题。

This chapter clarifies the nested relationship between the matrix family of unitary, orthogonal, Givens, Householder, permutation, and row or column symmetric matrices. A precise correspondence of the singular values and singular vectors between the unitary-symmetric matrix and its mother matrix is derived and proved (hence a fast algorithm of singular value decomposition for unitary-symmetric matrix is straightforwardly obtained), and the corresponding perturbation bound is provided.

该章揭示了酉对称矩阵、正交对称矩阵、 Givens 对称矩阵、 Householder 对称矩阵、置换对称矩阵和行对称矩阵之间的逐级包含关系；推导并证明了，酉对称矩阵的奇异值和奇异向量与母矩阵的奇异值和奇异向量之间的定量关系，据此可得酉对称矩阵奇异值分解的快速算法；给出并证明了摄动矩阵的摄动界。

This thesis focuses on studying the matrix equa-tion problem systematically, and proposed an abstract algorithm of solving the matrixequation with constraints, and established a strict convergence theory. Using this algo-rithm, we can solve the sets of matrix equation satisfying some constraint conditions,such as symmetric, antisymmetric, centrosymmetric, centroskew symmetric, re?exive,antire?exive, bisymmetric, symmetric and antipersymmetric, symmetric orthogonalsymmetric, symmetric orthogonal antisymmetric, Hermite generalized Hamilton ma-trix;So we can solve the problem with this algorithm, if the set of constrain matrixcan make a subspace in matrix space, and this algorithm also can solve the optimalapproximation and least squares problem. So this abstract algorithm has universal andimportant practical value.

本篇硕士论文系统地研究了此类问题，并找到了求解约束矩阵问题的抽象算法，并建立严格的收敛性理论，利用这一算法可求解约束条件为对称矩阵、反对称矩阵、中心对称矩阵、中心反对称矩阵、自反矩阵、反自反矩阵，对称正交对称矩阵、对称正交反对称矩阵、双中心矩阵、Hermite广义Hamilton矩阵等；可以说只要约束矩阵集合在矩阵空间中构成子空间，都可以考虑用此算法求解，而且这一算法还能把矩阵方程解及其最佳逼近，最小二乘解及其最佳逼近统一处理，因此本文算法有普适性和重要的实用价值。

The solutions of ProblemⅠ,ⅡandⅢare discussed by using the generalized conjugate gradient method. When the equation is consistent, the solutions such as symmetric, skew-symmetric, centrosymmetric, centroskew symmetric, reflexive, antireflexive, bisymmetric or symmetric and antipersymmetric are successfully found; When the equation is inconsistent, the least-squares solutions such as symmetric, skew-symmetric, centrosymmetric, centroskew symmetric, reflexive, antireflexive, bisymmetric or symmetric and antipersymmetric are also found successfully. The generalized conjugate gradient method has the following traits:(1) It can judge automatically the information of solutions.

利用广义共轭梯度法，讨论了问题Ⅰ、Ⅱ和Ⅲ解的情况：当方程相容时，研究了方程的一般解、对称解、中心对称解、自反矩阵解、双对称解、对称次反对称解及其最佳逼近等问题；当方程不相容时，研究了方程的最小二乘一般解、最小二乘对称解、最小二乘中心对称解、最小二乘自反矩阵解、最小二乘双对称解、最小二乘对称次反对称解及其最佳逼近等问题。

Thesis and mainly discuss the following problems:What we mainly discussed in the second chapter as follows:(1) S1,S2 are sets of symmetric orth-symmetric matrices;(2) S1,S2 are sets of bisymmetric matrices;(3) S1,S2 are sets of anti-symmetric orth-anti-symmetric matrices;(4) S1,S2 are sets of bi-anti-symmetric matrices;(5) S1 is the set of symmetric orth-symmetric matrices, S2 is the set of anti-symmetric orth-anti-symmetric matrices;(6) S1 is the set of bisymmetric matrices, S2 is the set of bi-anti-symmetric matrices;(7) S1 is the set of anti-symmetric orth-anti-symmetric matrices, S2 is the set of symmetric orth-symmetric matrices;(8) S1 is the set of bi-anti-symmetric matrices, S2 is the set of bisymmetricmatrices;On the base of studying the basic properties of the matrices, the expression of solutions and some numerical examples are presented.

本文第二章将主要就上述问题讨论如下几种情况： 1.S_1，S_2为对称正交对称矩阵； 2.S_1，S_2为双对称矩阵； 3.S_1，S_2为反对称正交反对称矩阵； 4.S_1，S_2为双反对称矩阵； 5.S_1为对称正交对称矩阵，S_2为反对称正交反对称矩阵； 6.S_1为双对称矩阵，S_2为双反对称矩阵； 7.S_1为反对称正交反对称矩阵，S_2为对称正交对称矩阵； 8.S_1为双反对称矩阵，S_2为双对称矩阵。

Problem III Given find such thatProblem IV When Problem I or II or III is consistent, let Se denote the set of its solutions, for given , find , such thatwhere is Frobenius norm, S is Rn×p or a subset of Rn×p satisfying some constraint conditions, such as symmetric, skew-symmetric, centrosymmet-ric, centroskew symmetric, reflexive, antireflexive, bisymmetric or symmetric and antipersymmetric.

问题Ⅳ 设问题Ⅰ或Ⅱ或Ⅲ相容，且其解集合为SE，给定X0∈Rn×p，求X∈SE，使其中‖·‖为Frobenius范数，S为Rn×p或为Rn×p中满足某约束条件的矩阵集合，如对称矩阵、反对称矩阵、中心对称矩阵、中心反对称矩阵、自反矩阵、反自反矩阵、双对称矩阵、对称次反对称矩阵等。

R~, find∈S_E, such that ProblemⅤGiven, find [X_1,X_2,…,X_l](where X_i∈S_i,i=1,2,…,l), such that A_1X_1B_1+A_2X_2B_2+…+A_lX_lB_l=C ProblemⅥWhen ProblemⅤis consistent, let SE denote the set of its solutions, for given,find, such that where||·|| is Frobenius norm, S and S_i are the matrix set satisfying some constraint conditions such as symmetric, skew-symmetric, centrosymmetric, centroskew symmetric, reflexive, antireflexive, bisymmetric or symmetric and antipersymmetric.

R~，求∈S_E，使得问题Ⅴ给定，求[X_1，X_2，…，X_l](其中X_i∈S_i，i=1，2，…，l)，使得 A_1X_1B_1+A_2X_2B_2+…+A_lX_lB_l=C 问题Ⅵ设问题Ⅴ相容，且其解集合为S_E，给定矩阵组，求，使得其中||·||为Frobenius范数，S，S_i为满足某种约束条件的矩阵集合，如对称矩阵、反对称矩阵、中心对称矩阵、中心反对称矩阵、自反矩阵、反自反矩阵、双对称矩阵、对称次反对称矩阵等等。

Denotes the Frobenius norm, S is a subset of Rn×n. This master thesishas mainly studied centrosymmetric matrix set, centroskew symmetric matrix set,re?

为Frobenius范数， S为Rn×n中满足某约束条件的矩阵集合，本硕士论文主要研究了中心对称矩阵、中心反对称矩阵、自反矩阵、反自反矩阵、双对称矩阵、对称次反对称矩阵、对称正交对称矩阵、对称正交反对称矩阵。

For problem II, many references have studied it and obtained its common solutions, symmetric solutions, skew-symmetric solutions and its optimal approximation constrained solution, but the representation of its solutions are complicated.

本文首次采用迭代法系统的研究了它求一般解、对称解、反求解约束矩阵方程及其最佳逼近的迭代法的研究对称解、中心对称解、中心反对称解、自反矩阵解、反自反矩阵解、双对称解、对称次反对称解及其最佳逼近问题，并首次成功地解决了它求中心对称解、中心反对称解、自反矩阵解、反自反矩阵解、双对称解与对称次反对称解及其最佳逼近的问题，拓广和改进了已有的研究成果。

The absorption and distribution of chromium were studied in ryeusing nutrient culture technique and pot experiment.

采用不同浓度K2CrO4(0，0.4，0.8和1.2 mmol/L)的Hoagland营养液处理黑麦幼苗，测定铬在黑麦体内的亚细胞分布、铬化学形态及不同部位的积累。

By analyzing theory foundation of mathematical morphology in the digital image processing, researching morphology arithmetic of the binary Image, discussing two basic forms for the least structure element: dilation and erosion.

通过分析数学形态学在图像中的理论基础，研究二值图像的形态分析算法，探讨最小结构元素的两种基本形态：膨胀和腐蚀；分析了数学形态学复杂算法的基本原理，把数学形态学的部分并行处理理念引入到家实际应用中。