# 数学代写|离散数学作业代写discrete mathematics代考|Math1030Q

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## 数学代写|离散数学作业代写discrete mathematics代考|Matrix Operations

More general sets of linear equations may be solved with $m \times n$ matrices (i.e. a matrix with $m$ rows and $n$ columns) or square $n \times n$ matrices. In this section, we consider several matrix operations including addition, subtraction, multiplication of matrices, scalar multiplication and the transpose of a matrix.

The addition and subtraction of two matrices $\mathrm{A}, \mathrm{B}$ is meaningful if and only if $\mathrm{A}$ and $\mathrm{B}$ have the same dimensions: i.e. they are both $m \times n$ matrices. In this case, $\mathrm{A}+\mathrm{B}$ is defined by adding the corresponding entries:
$$(\mathrm{A}+\mathrm{B}){i j}=\mathrm{A}{i j}+\mathrm{B}{i j}$$ The additive identity matrix for the square $n \times n$ matrices is denoted by 0 , where 0 is a $n \times n$ matrix whose entries are zero: i.e. $r{i j}=0$ for all $i, j$ where $1 \leq i \leq n$ and $1 \leq j \leq n$

The scalar multiplication of a matrix A by a scalar $k$ is meaningful and the resulting matrix $k \mathrm{~A}$ is given by
$$(k A){i j}=k \mathbf{A}{i j} .$$
The multiplication of two matrices $\mathrm{A}$ and $\mathrm{B}$ is meaningful if and only if the number of columns of $\mathrm{A}$ is equal to the number of rows of $\mathrm{B}$ (Fig. 8.2): i.e. $\mathrm{A}$ is an $m \times n$ matrix and $\mathrm{B}$ is a $n \times p$ matrix and the resulting matrix $\mathrm{AB}$ is a $m \times p$ matrix.

Let $\mathrm{A}=\left(a_{i j}\right)$ where $i$ ranges from 1 to $m$ and $j$ ranges from 1 to $n$, and let $\mathrm{B}=\left(b_{j l}\right)$ where $j$ ranges from 1 to $n$ and $l$ ranges from 1 to $p$. Then, $\mathrm{AB}$ is given by $\left(c_{i l}\right)$ where $i$ ranges from 1 to $m$ and $l$ ranges from 1 to $p$ with $c_{i l}$ given by
$$c_{i l}=\sum_{k=1}^n a_{i k} b_{k l}$$
That is, the entry $\left(c_{i l}\right)$ is given by multiplying the $t^{\text {th }}$ row in A by the $l^{\text {th }}$ column in $B$ followed by a summation. Matrix multiplication is not commutative: i.e. $\mathrm{AB} \neq \mathrm{BA}$

## 数学代写|离散数学作业代写discrete mathematics代考|Eigen Vectors and Values

A number $\lambda$ is an eigenvalue of a $n \times n$ matrix A if there is a non-zero vector $v$ such that the following equation holds:
$$\mathrm{A} v=\lambda v .$$
The vector $v$ is termed an eigenvector and the equation is equivalent to
$$(\mathrm{A}-\lambda \mathrm{I}) v=0 .$$
This means that $(\mathrm{A}-\lambda \mathrm{I})$ is a zero divisor and hence it is not an invertible matrix. Therefore,
$$\operatorname{det}(\mathrm{A}-\lambda \mathrm{I})=0 .$$

The polynomial function $p(\lambda)=\operatorname{det}(\mathrm{A}-\lambda \mathrm{I})$ is called the characteristic polynomial of $\mathrm{A}$, and it is of degree $n$. The characteristic equation is $p(\lambda)=0$ and as the polynomial is of degree $n$ there are at most $n$ roots of the characteristic equation, and so there at most $n$ eigenvalues.

The Cayley-Hamilton theorem states that every matrix satisfies its characteristic equation: i.e. the application of the characteristic polynomial to the matrix A yields the zero matrix.
$$p(\mathrm{~A})=0$$

# 离散数学代写

## 数学代写|离散数学作业代写离散数学代考|矩阵运算

$$(\mathrm{A}+\mathrm{B}){i j}=\mathrm{A}{i j}+\mathrm{B}{i j}$$ 平方的加性单位矩阵 $n \times n$ 矩阵用0表示，其中0是a $n \times n$ 矩阵的项为零:即。 $r{i j}=0$ 为所有人 $i, j$ 哪里 $1 \leq i \leq n$ 和 $1 \leq j \leq n$

$$(k A){i j}=k \mathbf{A}{i j} .$$

$$c_{i l}=\sum_{k=1}^n a_{i k} b_{k l}$$

## 数学代写|离散数学作业代写离散数学代考|特征向量和值

$$\mathrm{A} v=\lambda v .$$

$$(\mathrm{A}-\lambda \mathrm{I}) v=0 .$$

$$\operatorname{det}(\mathrm{A}-\lambda \mathrm{I})=0 .$$

$$p(\mathrm{~A})=0$$

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## 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。

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