矩阵范数

And then, this paper presents an algorithm of computing the left inverse or right inverse for these special rectangle matrices. In thenormal algorithms for solving these problems, we need O(m2n) multiplications ordivisions. The algorithms in this paper only need O + O(n2) multiplications ordivisions. The paper is built as follows.

常规的计算m×n阶不相容线性方程组的极小范数最小二乘解的算法所需计算量为O(m~2n)，而本文根据特殊矩阵的特殊结构，通过两种不同的构造方式给出了两类不同的计算极小范数最小二乘解的快速算法，并给出了求特殊矩阵的左逆及右逆的快速算法，它们的计算量均为O+O(n~2)。

The matrix operator norm and unitarily invariant norm are two important normclasses.

0引言矩阵的算子范数和矩阵的西不变范数是两大类不同的重要范数类。

This paper summarizes the theory foundation and implementation of federated Kalman filter,and proposes a new dynamic informationsharing algorithm based on matrix norm,and designs a multinavigation system consisting of INS,GPS and Doppler.

总结了联邦滤波器理论的基础和实现方法，提出了一种基于矩阵范数进行信息动态分配的联邦滤波新算法。

The relationship between selecting range of controlled weighted factor and state matrix norm of discrete state equation is proposed when multi-step predictive selftuning control algorithm is adopted.

4提出了多步预测的自校正控制GPC算法控制加权因子的选择范围与离散状态方程状态矩阵范数之间的关系，给出了其它参数的选择方法，通过不同采样周期条件下仿真和实验研究与理论计算结果是一致。

Some methods for dealing with the stability problem has presented and these methods include: Lyapunov function, differential inequality, matrix norm and measure, Razumikhin theorem method etc. The results of stability for these systems are derived by using these methods. This paper may be summarized as follows.

本文考察了区间Lurie型控制系统与不确定时滞系统的稳定性问题，给出了判定系统稳定的一些处理方法，这些方法包括：Lyapunov函数法、微分不等式法、矩阵范数与测度法、Razumikhin定理等方法，并且应用这些方法得到了系统稳定的判别条件。

In Chapter 5, some knowledge about large-scale system and methods for investigating them are introduced, Lyapunov function, differential inequality, Razumikhin theorem, matrix norm and measure methods have been used to discuss the stability of uncertain large-scale system with delays and decay estimates for uncertain time-varying large-scale system with delays, the criteria of stability and decay estimates for uncertain large-scale system with delays are given.

第五章首先介绍了大系统的有关知识以及时滞大系统研究的方法，然后利用Lyapunov函数法、微分不等式法、Razumikhin定理法、矩阵范数与测度法研究了不确定定常时滞大系统的稳定性以及不确定时变时滞大系统的稳定性与系统衰减率的估计问题，得到了系统稳定的判别条件以及系统衰减率的估计。

On the basis of the definition of matrix traces , this paper discusses their characteristics at first and then according to the norm of the F of square matrix and Cauchy-Schwarz inequality gives how to prove the zero matrix, unsimilar matrix, number cloth matrix, column matrix idempotent matrix and non-equality matrix.

根据矩阵迹的定义，首先给出了矩阵迹的性质，然后依据方阵的F—范数定义Cauchy—Schwarz不等式，给出了零矩阵，不相似矩阵，数幂矩阵，列矩阵，幂等矩阵及矩阵不等式的证法。

A numerical algorithm and a numerical experiment are given. 2. The least norm, positive semidefinite and positive definite real solutions of bisymmetric matrix equations (A~TXA, XA — YAD)=(D, 0)are considered by applying the generalized singular value decomposition.

利用广义奇异值分解讨论了矩阵方程(A~T XA，XA-YAE)=(E，0)在对称次反对称矩阵集合中的解、极小范数解和在双对称矩阵集合中的解、极小范数解及正定解； 3。

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。

In the last of this dissertation, we give necessary and sufficient conditions for the existence of and the expressions of perpositive semidefinite solutions and bipositive semidefinite solutions to some systems of quaternion matrix equations.

给出了某些四元数矩阵方程组有广亚半正定解，双半正定解的充要条件及其表达公式；研究了矩阵方程组的最小二乘解，极小范数的最小二乘解，在某些约束条件下的最小二乘解和极小范数的最小二乘解，给出了其有这些解的充要条件及其表达公式。

According to the clear water experiment, aeration performance of the new equipment is good with high total oxygen transfer coefficient and oxygen utilization ratio.

曝气设备的动力效率在叶轮转速为120rpm～150rpm时取得最大值，此时氧利用率和充氧能力也具有较高值。

The environmental stability of that world - including its crushing pressures and icy darkness - means that some of its most famous inhabitants have survived for eons as evolutionary throwbacks, their bodies undergoing little change.

稳定的海底环境─包括能把人压扁的压力和冰冷的黑暗─意谓海底某些最知名的栖居生物已以演化返祖的样态活了万世，形体几无变化。