# 线性代数作业代写linear algebra代考|Identifying second degree equations

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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|The eigenvalue method

In this section we apply eigenvalue methods to determine the geometrical nature of the second degree equation
$$a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0,$$
where not all of $a, h, b$ are zero.
Let $A=\left[\begin{array}{ll}a & h \ h & b\end{array}\right]$ be the matrix of the quadratic form $a x^{2}+2 h x y+b y^{2}$. We saw in section 6.1, equation $6.2$ that $A$ has real eigenvalues $\lambda_{1}$ and $\lambda_{2}$, given by
$$\lambda_{1}=\frac{a+b-\sqrt{(a-b)^{2}+4 h^{2}}}{2}, \lambda_{2}=\frac{a+b+\sqrt{(a-b)^{2}+4 h^{2}}}{2} .$$
We show that it is always possible to rotate the $x, y$ axes to $x_{1}, x_{2}$ axes whose positive directions are determined by eigenvectors $X_{1}$ and $X_{2}$ corresponding to $\lambda_{1}$ and $\lambda_{2}$ in such a way that relative to the $x_{1}, y_{1}$ axes, equation $7.1$ takes the form
$$a^{\prime} x^{2}+b^{\prime} y^{2}+2 g^{\prime} x+2 f^{\prime} y+c=0 .$$
Then by completing the square and suitably translating the $x_{1}, y_{1}$ axes, to new $x_{2}, y_{2}$ axes, equation $7.2$ can be reduced to one of several standard forms, each of which is easy to sketch. We need some preliminary definitions.

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

An $n \times n$ real matrix $P$ is called orthogonal if
$$P^{t} P=I_{n} .$$
It follows that if $P$ is orthogonal, then $\operatorname{det} P=\pm 1$. For
$$\operatorname{det}\left(P^{t} P\right)=\operatorname{det} P^{t} \operatorname{det} P=(\operatorname{det} P)^{2},$$
so $(\operatorname{det} P)^{2}=\operatorname{det} I_{n}=1$. Hence $\operatorname{det} P=\pm 1$.
If $P$ is an orthogonal matrix with $\operatorname{det} P=1$, then $P$ is called a proper orthogonal matrix.

THEOREM 7.1.1 If $P$ is a $2 \times 2$ orthogonal matrix with $\operatorname{det} P=1$, then
$$P=\left[\begin{array}{rr} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{array}\right]$$
for some $\theta$.
REMARK 7.1.1 Hence, by the discusssion at the beginning of Chapter 6 , if $P$ is a proper orthogonal matrix, the coordinate transformation
$$\left[\begin{array}{l} x \ y \end{array}\right]=P\left[\begin{array}{l} x_{1} \ y_{1} \end{array}\right]$$
represents a rotation of the axes, with new $x_{1}$ and $y_{1}$ axes given by the repective columns of $P$.

## 线性代数作业代写linear algebra代考|The eigenvalue method

λ1=一种+b−(一种−b)2+4H22,λ2=一种+b+(一种−b)2+4H22.

[X 和]=磷[X1 和1]

# 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