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# 统计代写|线性模型代写Linear Model代考|STAT393 Definition and existence

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## 统计代写|线性模型代写Linear Model代考|Definition and existence

A generalized inverse of a matrix $\mathbf{A}$ is defined, in this book, as any matrix $\mathbf{G}$ that satisfies the equation
$$\mathbf{A} \mathbf{G A}=\mathbf{A} .$$
The name “generalized inverse” for matrices $\mathbf{G}$ defined by (1) is unfortunately not universally accepted, although it is used quite widely. Names such as “conditional inverse”, “pseudo inverse” and ” $g$-inverse” are also to be found in the literature, sometimes for matrices defined as is $\mathbf{G}$ of (1) and sometimes for matrices defined as variants of $\mathbf{G}$. However, throughout this book the name “generalized inverse” of $\mathbf{A}$ is used exclusively for any matrix $\mathbf{G}$ satisfying (1).

Notice that (1) does not define $\mathbf{G}$ as “the” generalized inverse of $\mathbf{A}$ but as “a” generalized inverse. This is because $\mathbf{G}$, for a given matrix $\mathbf{A}$, is not unique. As shown below, there is an infinite number of matrices $\mathbf{G}$ that satisfy (1) and so we refer to the whole class of them as generalized inverses of $\mathbf{A}$.

One way of illustrating the existence of $\mathbf{G}$ and its non-uniqueness starts with the equivalent diagonal form of $\mathbf{A}$. If $\mathbf{A}$ has order $p \times q$ the reduction to this diagonal form can be written as
$$\mathbf{P}{p \times p} \mathbf{A}{p \times Q} \mathbf{Q}{q \times Q}=\Delta{p \times a} \equiv\left[\begin{array}{ll} \mathbf{D}{r \times r} & \mathbf{0}{r \times(q-r)} \ \mathbf{0}{(p-r) \times r} & \mathbf{0}{(p-r) \times(a-r)} \end{array}\right]$$
or, more simply, as
$$\mathbf{P A Q}=\Delta=\left[\begin{array}{cc} \mathbf{D}_{r} & 0 \ 0 & 0 \end{array}\right]$$

## 统计代写|线性模型代写Linear Model代考|Consistent equations

A convenient starting point from which to develop the solution of linear equations using a generalized inverse is the definition of consistent equations.
Definition. The linear equations $\mathbf{A x}=\mathbf{y}$ are defined as being consistent if any linear relationships existing among the rows of $\mathbf{A}$ also exist among the corresponding elements of $\mathbf{y}$.

As a simple example, the equations
$$\left[\begin{array}{ll} 1 & 2 \ 3 & 6 \end{array}\right]\left[\begin{array}{l} x_{1} \ x_{2} \end{array}\right]=\left[\begin{array}{r} 7 \ 21 \end{array}\right]$$
are consistent: in the matrix on the left the second row is thrice the first, and this is also true of the elements on the right. But the equations
$$\left[\begin{array}{ll} 1 & 2 \ 3 & 6 \end{array}\right]\left[\begin{array}{l} x_{1} \ x_{2} \end{array}\right]=\left[\begin{array}{l} 7 \ 24 \end{array}\right]$$
are not consistent. Further evidence of this is seen by writing them in full:
$$x_{1}+2 x_{2}=7 \quad \text { and } \quad 3 x_{1}+6 x_{2}=24 .$$
As a consequence of the first, $3 x_{1}+6 x_{2}=21$, which cannot be true if the second is to hold. The equations are therefore said to be inconsistent.

## 统计代写|线性模型代写Linear Model代考|Definition and existence

$$\mathbf{A G A}=\mathbf{A} .$$

inverse”也可以在文献中找到，有时用于定义如下的矩阵 $\mathbf{G}$ (1) 的，有时对于定义为㚆体的矩阵 $\mathbf{G}$. 然而，在整本书中， “义逆” 这个名称 $\mathbf{A}$ 专门用于任何矩阵 $\mathbf{G}$ 满足 (1)。

$$\mathbf{P A Q}=\Delta=\left[\begin{array}{llll} \mathbf{D}{r} & 0 & 0 & 0 \end{array}\right]$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。