矩阵范数

By means of analyzing the noise error, it is proposed to use an algorithm of singular value decomposition total least squares to conduct the interharmonics estimation, namely taking the noise interference at both sides of the matrix equation into account simultaneously and solving the minimum norm problem by least square method under such condition. In the proposed algorithm by means of singular value decomposition of a new augmented matrix and by use of a simple and practical method to determine the number of principal singular value related to signal to noise ratio, the storage calculation of right singular matrix is performed, thus a more accurate frequency estimation of interharmonics can be obtained.

通过分析噪声误差，提出采用奇异值总体最小二乘(singular value decomposition total least squares，SVDTLS)算法进行间谐波频率估计，即同时考虑矩阵方程两边的噪声干扰，采用SVDTLS算法求解该情况下的最小范数解，通过对增广矩阵进行奇异值分解(singular value decomposition，SVD)，采用简单实用的与信噪比相关的主奇异值个数确定方法对分解的右奇异矩阵进行存储计算，得到了较精确的间谐波频率估计结果。

Then,a new fast algorithm of the minimal norm least squares solution for linear system whose coefficients is an m×n symmetric Loewner matrix with full column rank is given by forming a special block matrix and researching the triangular factorization of its inverse.

对于工程计算中常常遇到的一类线性方程组的求解，通过构造特殊分块矩阵并研究其逆矩阵的三角分解，给出了求秩为n的m×n阶对称Loewner矩阵为系数阵的线性方程组，及极小范数最小二乘解的快速算法，该算法的计算复杂度为O+O(n2)，而一般方法的计算复杂度为O(mn2)+O(n3)。

The new updating method is to minimize the error matrix norm under the condition of a quadratic constraint under which the mode orthogonality should be satisfied and the mass and stiffness matrices should be fitted into the eigen equation.

该方法是在质量矩阵和刚度矩阵满足正交性条件和特征方程的约束下，使修正矩阵的范数最小，将模型修正问题转化为一个带二次约束的最小二乘问题。

The corresponding least-squares solution with minimum norm can be also obtained by choosing a kind of special initial matrices.

在不考虑舍入误差的情况下，对任意的初始矩阵都可以在有限步计算出它们在给定矩阵集合中的一个最小二乘解，若选取特殊的初始矩阵，则可以得到相应的最小范数最小二乘解。

From the main results, when the spectral norm of the system matrix is less than 1 or the matrix induced by the system matrix is Schur, the fuzzy system is asymptotically stable.

主要结论指出，当系统矩阵谱范数小于1或系统矩阵的诱导矩阵是Schur矩阵时，模糊系统全局渐近稳定。

The method of estimation in paper [7] is to build fitness function by the 2-norm of error matrix directly. For the error matrix is a Toeplitz matrix, a novel fitness function was presented. Then a new method, the real-valued genetic algorithm which is based on genes weights variation independently, is developed in this paper.

文献[7]提出的估计方法直接以误差矩阵的2范数建立适应度函数，文中利用该矩阵是Toeplitz矩阵的特性，构造了一个新的适应度函数，提出了一种基于基因分量独立变异的实数遗传算法。

For any initial generalized bisymmetric matrix $X_1$, when $AXB=C$ is consistent, we can obtain the generalized bisymmetric solution of the matrix equation AXB=C within finite iterative steps by the iteration method in the absence of roundoff errors; Moreover, the least-norm solution $X^*$ can be obtained by choosing a special kind of initial generalized bisymmetric matrix.

在不考虑机器误差的情况下，当矩阵方程AXB=C相容时，对任意广义双对称X_1，矩阵方程AXB=C的解可以经过有限步迭代得到；特别地，通过选择特殊地初始广义双对称矩阵极小范数解X^*。

Furthermore, its optimal approximation to a given matrix X can be obtained by finding the least-norm centrosymmetric solution of a new matrix equation AXB+CXD=F.

对任意给定的矩阵Xo，矩阵方程AXB+CXD=F的最佳逼近中心对称解可以通过迭代求解新的矩阵方程AXB+CXD=F的极小范数中心对称解而得到。

By this iterative method ,the solvability of the matrix equation can be judged automatically.When the matrix equation is consistent ,then,for any initial centrosymmetric matrix pair ,a centrosymmetric solution pair can be obtained within finite iteration steps in the absence of roundoff errors, and the least norm centrosymmetric solution pair can be obtained by choosing a special kind of initial centrosymmetric matrix pair.

当矩阵方程相容时，对任意初始的中心对称矩阵对在没有舍入误差的情况下，经过有限步迭代，得到它的一个中心对称解，并且，通过选择一种特殊的中心对称矩阵对，得到它的最小范数中心对称解。

When the matrix equation is consistent, its centrosymmetric solution can be obtained within finite iterative steps in the absence of round off errors for any initial centrosymmetric matrix X1, and its least-norm centrosymmetric solution can be derived by choosing a suitable initial iterative matrix.

当矩阵方程AXB+CXD=F有中心对称解时，在有限的误差范围内，对任意初始中心对称矩阵X1，运用迭代算法，经过有限步可得到矩阵方程的中心对称解；选取合适的初始迭代矩阵，还可以迭代出极小范数中心对称解。

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。