# 线性代数网课代修|数值线性代数代写Numerical linear algebra代考|STAT24300

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• 数值分析
• 高等线性代数
• 矩阵论
• 优化理论
• 线性规划
• 逼近论

## 线性代数作业代写linear algebra代考|Determinants

The first systematic treatment of determinants was given by Cauchy in $1812 . \mathrm{He}$ adopted the word “determinant”. The first use of determinants was made by Leibniz in 1693 in a letter to De L’Hôspital. By the beginning of the twentieth century the theory of determinants filled four volumes of almost 2000 pages (Muir, 1906-1923. Historic references can be found in this work). The main use of determinants in this text will be to study the characteristic polynomial of a matrix and to show that a matrix is nonsingular.
For any $\boldsymbol{A} \in \mathbb{C}^{n \times n}$ the determinant of $\boldsymbol{A}$ is defined by the number
$$\operatorname{det}(\boldsymbol{A})=\sum_{\sigma \in S_{n}} \operatorname{sign}(\sigma) a_{\sigma(1), 1} a_{\sigma(2), 2} \cdots a_{\sigma(n), n}$$
This sum ranges of all $n$ ! permutations of ${1,2, \ldots, n}$. Moreover, $\operatorname{sign}(\sigma)$ equals the number of times a bigger integer precedes a smaller one in $\sigma$. We also denote the determinant by (Cayley, 1841)
$$\left|\begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1 n} \ a_{21} & a_{22} & \cdots & a_{2 n} \ \vdots & \vdots & & \vdots \ a_{n 1} & a_{n 2} & \cdots & a_{n n} \end{array}\right|$$
From the definition we have
$$\left|\begin{array}{ll} a_{11} & a_{12} \ a_{21} & a_{22} \end{array}\right|=a_{11} a_{22}-a_{21} a_{12} .$$

## 线性代数作业代写linear algebra代考|Eigenvalues, Eigenvectors and Eigenpairs

Suppose $\boldsymbol{A} \in \mathbb{C}^{n \times n}$ is a square matrix, $\lambda \in \mathbb{C}$ and $\boldsymbol{x} \in \mathbb{C}^{n}$. We say that $(\lambda, \boldsymbol{x})$ is an eigenpair for $\boldsymbol{A}$ if $\boldsymbol{A} \boldsymbol{x}=\lambda \boldsymbol{x}$ and $\boldsymbol{x}$ is nonzero. The scalar $\lambda$ is called an eigenvalue and $\boldsymbol{x}$ is said to be an eigenvector. ${ }^{1}$ The set of eigenvalues is called the spectrum of $\boldsymbol{A}$ and is denoted by $\sigma(\boldsymbol{A})$. For example, $\sigma(\boldsymbol{I})={1, \ldots, 1}={1}$.
Eigenvalues are the roots of the characteristic polynomial.
Lemma 1.5 (Characteristic Equation) For any $A \in \mathbb{C}^{n \times n}$ we have $\lambda \in$ $\sigma(\boldsymbol{A}) \Longleftrightarrow \operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})=0$

Proof Suppose $(\lambda, \boldsymbol{x})$ is an eigenpair for $\boldsymbol{A}$. The equation $A \boldsymbol{x}=\lambda x$ can be written $(\boldsymbol{A}-\lambda \boldsymbol{I}) \boldsymbol{x}=\mathbf{0}$. Since $\boldsymbol{x}$ is nonzero the matrix $A-\lambda \boldsymbol{I}$ must be singular with a zero determinant. Conversely, if $\operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})=0$ then $\boldsymbol{A}-\lambda \boldsymbol{I}$ is singular and $(A-\lambda I) x=0$ for some nonzero $x \in \mathbb{C}^{n}$. Thus $A x=\lambda x$ and $(\lambda, x)$ is an eigenpair for $\boldsymbol{A}$.

The expression $\operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})$ is a polynomial of exact degree $n$ in $\lambda$. For $n=3$ we have
$$\operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})=\left|\begin{array}{ccc} a_{11}-\lambda & a_{12} & a_{13} \ a_{21} & a_{22}-\lambda & a_{23} \ a_{31} & a_{32} & a_{33}-\lambda \end{array}\right| .$$

## 线性代数作业代写linear algebra代考|Determinants

Cauchy 在1812. He采用了“决定因素”一词。1693 年莱布尼茨在给 De L’Hôspital 的一封信 中首次使用了行列式。到 20 世纪初，行列式理论占据了将近 2000 页的四卷本（缪尔， 1906-1923 年。可以在这部著作中找到历史参考资料）。本文中行列式的主要用途是研究 矩阵的特征多项式并证明矩阵是非奇异的。

$$\operatorname{det}(\boldsymbol{A})=\sum_{\sigma \in S_{n}} \operatorname{sign}(\sigma) a_{\sigma(1), 1} a_{\sigma(2), 2} \cdots a_{\sigma(n), n}$$

$$\left|\begin{array}{llll} a_{11} & a_{12} & a_{21} & a_{22} \end{array}\right|=a_{11} a_{22}-a_{21} a_{12} .$$

## 线性代数作业代写linear algebra代考|Eigenvalues, Eigenvectors and Eigenpairs

$\boldsymbol{A} \boldsymbol{x}=\lambda \boldsymbol{x}$ 和 $\boldsymbol{x}$ 是非零的。标量 $\lambda$ 被称为特征值并且 $\boldsymbol{x}$ 被称为特征向量。1特征值的集合称为

# 计量经济学代写

## 在这种情况下，如何学好线性代数？如何保证线性代数能获得高分呢？

1.1 mark on book

【重点的误解】划重点不是书上粗体，更不是每个定义，线代概念这么多，很多朋友强迫症似的把每个定义整整齐齐用荧光笔标出来，然后整本书都是重点，那期末怎么复习呀。我认为需要标出的重点为

A. 不懂，或是生涩，或是不熟悉的部分。这点很重要，有的定义浅显，但证明方法很奇怪。我会将晦涩的定义，证明方法标出。在看书时，所有例题将答案遮住，自己做，卡住了就说明不熟悉这个例题的方法，也标出。

B. 老师课上总结或强调的部分。这个没啥好讲的，跟着老师走就对了

C. 你自己做题过程中，发现模糊的知识点

1.2 take note

1.3 understand the relation between definitions