矩阵范数

Chapter 1 mainly introduce some elementary concepts about matrix: consistently ordered matrix, matrix norm, nonnegative matrix and famous Perron-Frobenius theorem.

介绍了矩阵的一些基本概念：相容次序矩阵、矩阵范数、非负矩阵及著名的Perron-Frobenius定理等。

By using matrix norm, matrix singular value, non-negative matrix theory, the regions of numaerical range and spectum of operator polynomial and the regularity of operator linear pencil are obtained.

8应用矩阵范数理论、矩阵的奇异值理论、非负矩阵的理论、友矩阵、算子的谱理论为工具，获得了算子多项式数值域具有的特性及其包含范围，算子多项式数值域与n-次数值域的关系，最后研究了算子线性束正则性的等价条件及谱分布情况。

By the definition and properties of matrix norm, a corollary expressed by matrix norm is obtained.

又利用矩阵范数的定义及性质，得出了一个以范数形式表示的推论。

It is proved by means of constructing the new norm of the vector and the norm of matrix that the Runge-Kutta method is H-stable if|R | and A is nonsingular.

我们构造了一个新的向量范数和矩阵范数，利用该范数证明了如下定理，如果|R|和矩阵A非奇异，则Runge-Kutta方法是H-稳定的。

Inverse eigenvalue problem; Constrained matrix equation problem; Procrustes problem; Matrix norm; Optimal approximation solution

矩阵逆特征值问题；约束矩阵方程问题； Procrustes问题；矩阵范数；最佳逼近解

Keywords constrained matrix equation problem;sub matrix constraint problem; matrix norm; optimal approximation solution;least-squares solution

约束矩阵方程问题；子矩阵约束问题；矩阵范数；最佳逼近解；最小二乘解

In chapter5, the serial properties of numerical range of operator polynomial are put firstly. Secondly, the relation between numerical range of operator polynomial and n-numerical range are considered. Thirdly, in the light of matrix norm, matrix singular value, the regions and bounds of numerical range and spectrum of operator polynomial are dicussed carefully.

第五章研究了算子多项式数值域的性质、算子多项式数值域与n-次数值域的关系，特别地利用矩阵范数、矩阵的奇异值、非负矩阵的理论、友矩阵为工具，给出了算子多项式数值域及谱的范围全面刻画，并深刻地研究了算子线性束的正则性的等价条件及谱分布。

In this paper, we first show that the matrix2-norm is the only norm which is not only an operator norm but also a unitarily invariantnorm.

在［l，p.94］中曾提到，向量作为矩阵看待，归范化的西不变范数只有2一范数，本文将这一结论推广成，一个矩阵范数如果它既是算子范数又是规范化的西不变

Firstly,based on the B.Bowermans result about the rate of convergence in Cesaro sense of certain nonhomogeneous Markov chains which the transition matrices converge,we are to study a certain nonhomogenous Markov chains which the transition matrices average converge to a period strongly ergodic stochastic matrice,and control the average convergenc rate of transition matrices,then we get the rate of convergence in Cesaro sense about the nonhomogeneous Markov chains by used the character of norm and the character of nonhomogeneous Markov chains.It is an extension of a B.

首先在B.Bowerman等人研究转移矩阵列收敛的一类非齐次马氏链，其Cesaro平均收敛的收敛速度基础上，研究转移矩阵列平均收敛到一周期强遍历随机矩阵的一类非齐次马氏链，通过控制转移矩阵列平均收敛的收敛速度，利用矩阵范数的性质、非齐次马氏链的相关性质等，得到该非齐次马氏链转移矩阵Cesaro平均收敛的收敛速度，是B。

By comparing the matrix theory over number field,we mainly discussed the quaternion matrix norm theory,the quaternionic determinant,the complex number expression of the quaternion matrix and the quarternion linear matrix equation.

通过对比一般域上矩阵理论，本文主要从四元数矩阵范数理论、四元数矩阵行列式、四元数矩阵的复表示以及四元数线性方程组和线性矩阵方程四个方面阐述了四元数矩阵理论。

The split between the two groups can hardly be papered over.

这两个团体间的分歧难以掩饰。

This approach not only encourages a greater number of responses, but minimizes the likelihood of stale groupthink.

这种做法不仅鼓励了更多的反应，而且减少跟风的可能性。