矩阵的矩阵

Meantime we also get the explicit construction of pseudo-inverses of bordered matrix and obtain many new results.

本章所使用的主要工具是QQ-SVD，其中在研究加边矩阵{1，3}-和{1，4}-逆时，又使用了构造特殊非奇异矩阵的技巧，给出了加边矩阵{1，3}-和{1，4}-逆的显示结构，并进一步得到了加边矩阵伪逆的显示结构，获得了许多新的结果。

The program is the main description of the matrix operations, including the basic matrix operations, matrix inverse, matrix singular value decomposition, and the characteristic value and characteristic vector solution and so on.

详细说明：该程序是主要描述的是矩阵运算，包括矩阵的基本运算，矩阵的求逆，矩阵的奇异值分解，以及特征值以及特征向量的求解等等。

When scaled factor circulant matrices are nonsingular, we can find the single solution of the scaled factor circulant matrix equation; When scaled factor circulant matrices are singular, we can find the special as well as the general solution of the scaled factor circulant matrix equation; There is only an error of approximation when the fast algorithm is implemented on computers, and only the elements in the first row of the scaled factor circulant matrix and the constants in the diagonal matrix are needed by the fast algorithm.

当鳞状因子循环矩阵非奇异时，该快速算法求出线性方程组的唯一解；当鳞状因子循环矩阵奇异时，该快速算法求出线性方程组的特解与通解。该快速算法仅用到鳞状因子循环矩阵的第一行元素及对角矩阵中的对角上的常数进行计算。在计算机上实现时只有舍入误差。

This article is divided into three parts:I: It gives the relevant background knowledge, mainly about the study of circulant matrices at home and abroad, the basic concepts, characteristics of circulant matrices, and the basic computing instruments which have been frequently used in matrix theory and matrix calculations.

本文共分三个部分：一：给出相关的预备知识，主要是循环矩阵研究的国内外现状和进展、文中用到的循环矩阵的基本概念、性质以及在矩阵理论和矩阵计算中经常用到基本运算工具。

In Chapter three, by constructing a special partitioned matrix and studying the fast triangular factorization of its inverse matrix, we respectively give fast algorithms for the minimal norm least squares solutions of linear equations which coefficient matrices are n×m Toeplitz type matrices and Vandermonde-Loewner type matrices with full column rank.

第三章通过构造特殊分块矩阵并研究其逆矩阵的快速三角分解，分别给出了求以秩为m的n×m阶Toeplitz型矩阵和Vandermonde-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，并通过算例将新算法与已有的法方程组的方法和正交化方法作了比较。

The matrix analysis has already all kind of method. The back part of this document, according to the matrix's elementary operation, gives one matrix analyze method, this method can more earthly and convenient analyze the matrix, and also provide the real example of the value analysis.

矩阵的分解已经作出了各种方法，本文在后半部分中根据矩阵的初等变换，提出了一种分解矩阵的方法，该方法能够更通俗便捷的对矩阵进行分解，并用数值实例进行了分析。

One thing you have to take into consideration is that each call to such functions is equivalent to creating the corresponding translation, rotation or scaling matrix and then multiply the current model view matrix with this matrix (and storing the result in the model view matrix).

有一件事你必须要考虑的是，每个这种函数调用相当于创建相应的平移，旋转或缩放矩阵，然后乘以目前的模式与此矩阵视图矩阵（并存储在模型视图矩阵的结果）。

The conventional matrix decomposition methods have been used to find the least-squares solutions of the abovementioned inconsistent matrix equations over given matrix set in many references, and the general expressions of these solutions were obtained. But it is difficult to determine the solution of Problem I by utilizing these expressions due to the fact that the orthogonal invariance of Frobenius norm does no hold for the general nonsingular matrices.

对于求上述不相容矩阵方程在给定矩阵集合上的最小二乘解，很多文献中利用传统的矩阵分解方法得到了其通解表达式，但是利用该表达式很难得到问题Ⅰ的解，这是因为一般的非奇异矩阵并不满足Frobenius范数的正交不变性。

In the first part, we shall prove several inequalities involving symmetric positive semidefinite, general M-matrices and inverse M-matrice which are generalization of the classical Oppenheim s Inequality for symmetric positive semidefinite matrices.

第一个部分给出了半正定矩阵，一般的M-矩阵以及逆M-矩阵的一些相关不等式，而这些不等式都是有关半正定矩阵的经典的Oppenheim不等式的推广。

In the paper,we discuss orthogonal matrix deeply,and we get a series of properties of orthogonal matrix.

本文对正交矩阵进行了深入的研究，得到了正交矩阵的一系列性质，同时，本文对照正交矩阵的性质和运用，研究了两类广义正交矩阵的性质及应用。

I didn't watch TV last night, because it .

昨晚我没有看电视，因为电视机坏了。

Since this year, in a lot of villages of Beijing, TV of elevator liquid crystal was removed.

今年以来，在北京的很多小区里，电梯液晶电视被撤了下来。