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# 数学代写|数值分析代写Numerical analysis代考|MATH408 MATRIX ARITHMETIC

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## 数学代写|数值分析代写Numerical analysis代考|MATRIX ARITHMETIC

In working with linear systems, it is very useful to introduce arithmetic operations for the matrices that were introduced and used in the first two sections. Such matrix operations can simplify many derivations, and they lead to a conceptually clearer notation for writing linear systems of equations. In this section, the arithmetic of matrices will be studied and related to the solution of systems of linear equations.
A matrix is a rectangular array of numbers,
$$B=\left[\begin{array}{cccc} b_{11} & b_{12} & \ldots & b_{1 n} \ & \vdots & & \ b_{m 1} & b_{m 2} & \ldots & b_{m, n} \end{array}\right]$$
It is said to have order $m \times n$ if $m$ is the number of rows and $n$ is the number of columns. Square matrices with $n$ rows and $n$ columns are said to have order $n$. Matrices consisting of a single row or column are called row and column matrices, respectively. They are also called row or column vectors or, more simply, vectors. This is because they can be identified with geometric vectors drawn from the origin of a space to the point with coordinates given by the row or column matrix. This adds an important geometric perspective to the study of matrices, but we will omit it here.

## 数学代写|数值分析代写Numerical analysis代考|Arithmetic Operations

Let $B$ be the $m \times n$ matrix in (8.41) and let $a$ be an arbitrary real number. Then $a B$ is a matrix of order $m \times n$, defined by
$$a B=\left[\begin{array}{cccc} a b_{11} & a b_{12} & \cdots & a b_{1 n} \ \vdots & & & \ a b_{m 1} & a b_{m 2} & \cdots & a b_{m, n} \end{array}\right]$$
Each element of $B$ is multiplied by $a$ to get the corresponding element in $a B$.
Let $A$ and $B$ be matrices of order $m \times n$. Then $A+B$ is a new matrix of order $m \times n$, defined by
$$A+B=\left[\begin{array}{cccc} a_{11}+b_{11} & a_{12}+b_{12} & \cdots & a_{1 n}+b_{1 n} \ \vdots & & & \vdots \ a_{m 1}+b_{m 1} & a_{m 2}+b_{m 2} & \cdots & a_{m n}+b_{m n} \end{array}\right]$$
The $(i, j)$, element of $A+B$ is $a_{i j}+b_{i j}$. For matrix addition, there is a zero matrix. Define the zero matrix of order $m \times n$ as having all zero entries. It is denoted by $O_{m \times n}$ or, more commonly, $O$. It has the property that
$$A+O=O+A=A$$
for any matrix $A$.

## 数学代写|数值分析代写数值分析代考|MATRIX算术

$$B=\left[\begin{array}{cccc} b_{11} & b_{12} & \ldots & b_{1 n} \ & \vdots & & \ b_{m 1} & b_{m 2} & \ldots & b_{m, n} \end{array}\right]$$

## 数学代写|数值分析代写数值分析代考|算术运算

$$a B=\left[\begin{array}{cccc} a b_{11} & a b_{12} & \cdots & a b_{1 n} \ \vdots & & & \ a b_{m 1} & a b_{m 2} & \cdots & a b_{m, n} \end{array}\right]$$
$B$的每一个元素乘以$a$，得到$a B$中对应的元素

$$A+B=\left[\begin{array}{cccc} a_{11}+b_{11} & a_{12}+b_{12} & \cdots & a_{1 n}+b_{1 n} \ \vdots & & & \vdots \ a_{m 1}+b_{m 1} & a_{m 2}+b_{m 2} & \cdots & a_{m n}+b_{m n} \end{array}\right]$$

$$A+O=O+A=A$$
.

## 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。