### 数学代写|计算线性代数代写Computational Linear Algebra代考|MAST10007

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|计算线性代数代写Computational Linear Algebra代考|Eigenvalues, Eigenvectors and Eigenpairs

Suppose $A \in \mathbb{C}^{n \times n}$ is a square matrix, $\lambda \in \mathbb{C}$ and $\boldsymbol{x} \in \mathbb{C}^{n}$. We say that $(\lambda, x)$ 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 $A$ and is denoted by $\sigma(A)$. For example, $\sigma(I)={1, \ldots, 1}={1}$.
Eigenvalues are the roots of the characteristic polynomial.
Lemma $1.5$ (Characteristic Equation) For any $A \in \mathbb{C}^{n \times n}$ we have $\lambda \in$ $\sigma(A) \Longleftrightarrow \operatorname{det}(A-\lambda I)=0$

Proof Suppose $(\lambda, x)$ is an eigenpair for $\boldsymbol{A}$. The equation $A x=\lambda x$ can be written $(\boldsymbol{A}-\lambda \boldsymbol{I}) \boldsymbol{x}=\mathbf{0}$. Since $\boldsymbol{x}$ is nonzero the matrix $\boldsymbol{A}-\lambda \boldsymbol{I}$ 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
$$\operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})=\left|\begin{array}{ccc} a_{11}-\lambda & a_{12} & a_{13} \ a_{21} & a_{22}-\lambda & a_{23} \ a_{31} & a_{32} & a_{33}-\lambda \end{array}\right|$$
Expanding this determinant by the first column we find
\begin{aligned} \operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I}) &=\left(a_{11}-\lambda\right)\left|\begin{array}{cc} a_{22}-\lambda & a_{23} \ a_{32} & a_{33}-\lambda \end{array}\right|-a_{21}\left|\begin{array}{cc} a_{12} & a_{13} \ a_{32} & a_{33}-\lambda \end{array}\right| \ &+a_{31}\left|\begin{array}{cc} a_{12} & a_{13} \ a_{22}-\lambda & a_{23} \end{array}\right|=\left(a_{11}-\lambda\right)\left(a_{22}-\lambda\right)\left(a_{33}-\lambda\right)+r(\lambda) \end{aligned}
for some polynomial $r$ of degree at most one. In general
$$\operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})=\left(a_{11}-\lambda\right)\left(a_{22}-\lambda\right) \cdots\left(a_{n n}-\lambda\right)+r(\lambda),$$
where each term in $r(\lambda)$ has at most $n-2$ factors containing $\lambda$. It follows that $r$ is a polynomial of degree at most $n-2$, $\operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})$ is a polynomial of exact degree $n$ in $\lambda$ and the eigenvalues are the roots of this polynomial.

## 数学代写|计算线性代数代写Computational Linear Algebra代考|Exercises Sect

Exercise 1.1 (Strassen Multiplication (Exam Exercise 2017-1)) (By arithmetic operations we mean additions, subtractions, multiplications and divisions.)
Let $\boldsymbol{A}$ and $\boldsymbol{B}$ be $n \times n$ real matrices.
a) With $\boldsymbol{A}, \boldsymbol{B} \in \mathbb{R}^{n \times n}$, how many arithmetic operations are required to form the product $\boldsymbol{A B}$ ?
b) Consider the $2 n \times 2 n$ block matrix
$$\left[\begin{array}{ll} W & X \ Y & Z \end{array}\right]=\left[\begin{array}{ll} A & B \ C & D \end{array}\right]\left[\begin{array}{ll} E & F \ G & H \end{array}\right],$$
where all matrices $\boldsymbol{A}, \ldots, \boldsymbol{Z}$ are in $\mathbb{R}^{n \times n}$. How many operations does it take to compute $\boldsymbol{W}, \boldsymbol{X}, \boldsymbol{Y}$ and $\boldsymbol{Z}$ by the obvious algorithm?
c) An alternative method to compute $\boldsymbol{W}, \boldsymbol{X}, \boldsymbol{Y}$ and $\boldsymbol{Z}$ is to use Strassen’s formulas:
$\mathbf{P}{1}=(\boldsymbol{A}+\boldsymbol{D})(\boldsymbol{E}+\boldsymbol{H})$, $\mathbf{P}{2}=(\boldsymbol{C}+\boldsymbol{D}) \boldsymbol{E}, \quad \mathbf{P}{5}=(\boldsymbol{A}+\boldsymbol{B}) \boldsymbol{H}$, $\mathbf{P}{3}=\boldsymbol{A}(\boldsymbol{F}-\boldsymbol{H}), \quad \mathbf{P}{6}=(\boldsymbol{C}-\boldsymbol{A})(\boldsymbol{E}+\boldsymbol{F})$, $\mathbf{P}{4}=\boldsymbol{D}(\boldsymbol{G}-\boldsymbol{E}), \quad \mathbf{P}{7}=(\boldsymbol{B}-\boldsymbol{D})(\boldsymbol{G}+\boldsymbol{H})$, $\boldsymbol{W}=\mathbf{P}{1}+\mathbf{P}{4}-\mathbf{P}{5}+\mathbf{P}{7}, \quad \boldsymbol{X}=\mathbf{P}{3}+\mathbf{P}{5}$, $\boldsymbol{Y}=\mathbf{P}{2}+\mathbf{P}{4}, \quad \boldsymbol{Z}=\mathbf{P}{1}+\mathbf{P}{3}-\mathbf{P}{2}+\mathbf{P}{6} .$ You do not have to verify these formulas. What is the operation count for this method? d) Describe a recursive algorithm, based on Strassen’s formulas, which given two matrices $\boldsymbol{A}$ and $\boldsymbol{B}$ of size $m \times m$, with $m=2^{k}$ for some $k \geq 0$, calculates the product $\boldsymbol{A B}$. e) Show that the operation count of the recursive algorithm is $\mathcal{O}\left(m^{\log {2}(7)}\right)$. Note that $\log _{2}(7) \approx 2.8<3$, so this is less costly than straightforward matrix multiplication.

## 数学代写|计算线性代数代写Computational Linear Algebra代考|Cubic Spline Interpolation

Since there are $n+1$ interpolation conditions in (2.2) a natural choice for a function $g$ is a polynomial of degree $n$. As shown in most books on numerical methods such a $g$ is uniquely defined and there are good algorithms for computing it. Evidently, when $n=1, g$ is the straight line
$$g(x)=y_{1}+\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\left(x-x_{1}\right),$$
known as the linear interpolation polynomial.
Polynomial interpolation is an important technique which often gives good results, but the interpolant $g$ can have undesirable oscillations when $n$ is large. As an example, consider the function given by
$$f(x)=\arctan (10 x)+\pi / 2, \quad x \in[-1,1] .$$
The function $f$ and the polynomial $g$ of degree at most 13 satisfying (2.2) with $[a, b]=[-1,1]$ and $y_{i}=f\left(x_{i}\right), i=1, \ldots, 14$ is shown in Fig. 2.1. The interpolant has large oscillations near the end of the range. This is an example of the Runge phenomenon. Using larger $n$ will only make the oscillations bigger. ${ }^{\text {| }}$

## 数学代写|计算线性代数代写Computational Linear Algebra代考|Exercises Sect

a) 与一个,乙∈Rn×n, 需要多少次算术运算才能形成乘积一个乙?
b) 考虑2n×2n块矩阵

[在X 是从]=[一个乙 CD][和F GH],

c) 另一种计算方法在,X,是和从是使用 Strassen 的公式：

## 数学代写|计算线性代数代写Computational Linear Algebra代考|Cubic Spline Interpolation

G(X)=是1+是2−是1X2−X1(X−X1),

F(X)=反正切⁡(10X)+圆周率/2,X∈[−1,1].

## 有限元方法代写

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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。