查询词典 matrix norm

The least norm solutions of symmetric and skew antisymmetric matrix equations (A~TXA, XA — YAD)=(D, 0)are also considered. 3. The least-squares solutions and the least norm solutions of the quaternion matrix equations AX = B and XD = E are studied by using

借助四元数的复表示方法，研究四元数矩阵方程组AX=B，XD=E的最小二乘问题和极小范数解的问题，并对主子阵约束下的方程组的解进行了讨论，得到了它们的显式表示。

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。

In this paper, for m× n Loewner-type matrix or symmetric Loewner-type matrix with full column rank, we form special block matrix firstly and research their triangular factorization or of its inverse, and then get the fast algorithm of the minimal norm least squaresolution .

本文对于秩为n的m×n阶Loewner型或对称Loewner型矩阵L，通过构造特殊分块矩阵并研究其逆的三角分解或自身三角分解，进而得到了线性方程组Lx=b的极小范数最小二乘解快速算法，计算复杂度为O+O(n~2)，而一般方法的计算复杂度为O(mn~2)+O(n~3)。

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

通过构造特殊分块矩阵及其三角分解给出了求秩为n的m×n阶Loewner型矩阵为系数阵的线性方程组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)。

Fast algorithm of minimal norm least square solution for Cauchy linear system;2. For the m×n Cauchy-type matrix with full column rank,the fast algorithm of the minimal norm least square solution to the linear system Cx=b was obtained by construction of a special block matrix and study of the triangular decomposition.

对于秩为n的m×n阶Cauchy型矩阵C，通过构造特殊分块矩阵并研究其三角分解，进而得到了线性方程组C x=b的极小范数最小二乘解的快速算法，所需运算量为O+O(n2)，而通常构造法方程组的方法所需运算量为O(m n2)+O(n3)，用正交化法虽然避免了构造法方程组，但所需的运算量更大些。

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。

Furthermore,the optimal approximation bisymmetric solution pair to a given bisymmetric matrix pair in Frobenius norm can be obtained by finding the least norm bisymmetric solution pair of new matrix equation ,where .

另外，给定双对称矩阵对，通过求矩阵方程的双对称解对，得到它的最佳逼近双对称解对。

In addition, the unique optimal approximation solution $\\\\\\\\\\\\\\\\hat X$ to given matrix $X_0 $ in Frobenius norm can be derived by finding the least-norm generalized bisymmetric solution $\\\\\\\\\\\\\\\\widetilde X^\\\\\\\\\\\\\\\\ast$ of the new matrix equation $A\\\\\\\\\\\\\\\\widetilde X B=\\\\\\\\\\\\\\\\widetilde C$, here,$\\\\\\\\\\\\\\\\widetilde X=X-X_0$, and $\\\\\\\\\\\\\\\\widetilde C=C-AX_0B$.

另外，对于给定矩阵X_0的最佳逼近解$\\\\\\\\\\\\\\\\hat X$，可以通过求解新的矩阵方程$A\\\\\\\\\\\\\\\\widetilde X B=\\\\\\\\\\\\\\\\widetilde C$得到，其中$\\\\\\\\\\\\\\\\widetilde X=X-X_0$，$\\\\\\\\\\\\\\\\widetilde C=C-AX_0B$。

Furthermore, the optimal approximation centrosymmetric solution pair to a given centrosymmetric matrix pair in Frobenius norm can be obtained by finding the least norm centrosymmetric solution pair of new matrix equation ,where .

另外，给定中心对称矩阵对，通过求矩阵方程的中心对称解对，得到它的最佳逼近中心对称解对。

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。

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。