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# 数学代写|线性代数代写Linear algebra代考|Identity matrix

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## 数学代写|线性代数代写Linear algebra代考|Identity matrix

What does the term identity matrix mean?
The identity matrix is a matrix denoted by $\mathbf{I}$ such that
$$\mathbf{A I}=\mathbf{A} \text { for any matrix } \mathbf{A}$$
This is similar to real numbers where the number 1 is the identity element which satisfies $x 1=x$ for any real number $x$
What does the identity matrix look like?
It is a matrix with 1’s along the leading diagonal (top left to bottom right) and zeros elsewhere.
For example, $\left(\begin{array}{ll}1 & 0 \ 0 & 1\end{array}\right),\left(\begin{array}{lll}1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1\end{array}\right)$ and $\left(\begin{array}{llll}1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1\end{array}\right)$ are all identity matrices.
How can we distinguish between these 2 by 2, 3 by 3 and 4 by 4 identity matrices?
We can denote the size in the subscript of $I$ as $I_2, I_3$ and $I_4$ respectively.
Is the identity matrix, I, a square matrix?
Yes the identity must be a square matrix.

How can we write the formal definition of the identity matrix?
Definition (1.21). An identity matrix is a square matrix denoted by $\mathbf{I}$ and defined by
$$\mathbf{I}=\left(i_{k j}\right)=\left{\begin{array}{lll} 1 & \text { if } & k=j \ 0 & \text { if } & k \neq j \end{array}\right.$$
What does definition (1.21) mean?
All the entries in the leading diagonal (top left to bottom right) of a matrix I are 1, that is
$$i_{11}=i_{22}=i_{33}=i_{44}=\cdots=1$$
and all the other entries away from the leading diagonal are zero.
For a 2 by 2 matrix we have:
$$\left(\begin{array}{ll} a & b \ c & d \end{array}\right)\left(\begin{array}{ll} 1 & 0 \ 0 & 1 \end{array}\right)=\left(\begin{array}{ll} a & b \ c & d \end{array}\right) \text { where } a, b, c \text { and } d \text { are real numbers. }$$
The identity matrix is illustrated in the next example.

## 数学代写|线性代数代写Linear algebra代考|Inverse matrix

Let $x \neq 0$ be a real number then the inverse of $x$ is a real number $x^{-1}$ such that
$$x\left(x^{-1}\right)=1$$
1 is the identity element of real numbers.
What do you think an inverse matrix is?
Given a square matrix $\mathbf{A}$ then the inverse of $\mathbf{A}$ is a square matrix $\mathbf{B}$ such that
$$\mathbf{A B}=\mathbf{I}$$
where $\mathbf{I}$ is the identity matrix defined earlier.
The inverse matrix of $\mathbf{A}$ is denoted by $\mathbf{A}^{-1}$ where $\mathbf{A}^{-1}=\mathbf{B}$ in the above case. Note, that
$$\mathbf{A}^{-1} \neq \frac{1}{\mathbf{A}}$$
The inverse matrix is not equal to 1 over $\mathbf{A}$. Actually 1 over $\mathbf{A}$ is not defined for a matrix A.
We will define the process of finding $\mathbf{A}^{-1}$ later in this chapter.
Graphically, the inverse matrix $\mathbf{A}^{-1}$ performs the transformation shown in Fig. 1.39.
If a transformation $\mathbf{A}$ is applied to an object as shown in Fig. 1.39 then the transformation $\mathbf{A}^{-1}$ undoes $\mathbf{A}$ so the net result of $\mathbf{A}^{-1} \mathbf{A}=\mathbf{I}$ is to leave the object unchanged.
Definition (1.23). A square matrix $\mathbf{A}$ is said to be invertible or non-singular if there is a matrix $\mathbf{B}$ of the same size such that
$$\mathbf{A B}=\mathbf{B A}=\mathbf{I}$$
Matrix $\mathbf{B}$ is called the (multiplicative) inverse of $\mathbf{A}$ and is denoted by $\mathbf{A}^{-1}$.

## 数学代写|线性代数代写Linear algebra代考|Identity matrix

$$\mathbf{A I}=\mathbf{A} \text { for any matrix } \mathbf{A}$$

$$\mathbf{I}=\left(i_{k j}\right)=\left{\begin{array}{lll} 1 & \text { if } & k=j \ 0 & \text { if } & k \neq j \end{array}\right.$$

$$i_{11}=i_{22}=i_{33}=i_{44}=\cdots=1$$

$$\left(\begin{array}{ll} a & b \ c & d \end{array}\right)\left(\begin{array}{ll} 1 & 0 \ 0 & 1 \end{array}\right)=\left(\begin{array}{ll} a & b \ c & d \end{array}\right) \text { where } a, b, c \text { and } d \text { are real numbers. }$$

## 数学代写|线性代数代写Linear algebra代考|Inverse matrix

$$x\left(x^{-1}\right)=1$$
1是实数的单位元。

$$\mathbf{A B}=\mathbf{I}$$

$\mathbf{A}$的逆矩阵表示为$\mathbf{A}^{-1}$，其中在上述情况中$\mathbf{A}^{-1}=\mathbf{B}$。请注意
$$\mathbf{A}^{-1} \neq \frac{1}{\mathbf{A}}$$

$$\mathbf{A B}=\mathbf{B A}=\mathbf{I}$$

## 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。