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assignmentutor-lab™ 为您的留学生涯保驾护航 在代写凸优化Convex Optimization方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写凸优化Convex Optimization代写方面经验极为丰富，各种代写凸优化Convex Optimization相关的作业也就用不着说。

• 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|凸优化作业代写Convex Optimization代考|Singular value decomposition

The singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with many applications in signal processing and communications. Applications which employ the SVD include computation of pseudo-inverse, least-squares fitting of data, matrix approximation, and determination of rank, range space, and null space of a matrix and so on and so forth.
Let $\mathbf{A} \in \mathbb{R}^{m \times n}$ with $\operatorname{rank}(\mathbf{A})=r$. The SVD of $\mathbf{A}$ is expressed in the form
$$\mathbf{A}=\mathbf{U} \Sigma \mathbf{V}^{T} \text {. }$$
In the full SVD (1.107) for $\mathbf{A}, \mathbf{U}=\left[\mathbf{U}{r}, \mathbf{U}^{\prime}\right] \in \mathbb{R}^{m \times m}$ and $\mathbf{V}=\left[\mathbf{V}{r}, \mathbf{V}^{\prime}\right] \in \mathbb{R}^{n \times n}$ are orthogonal matrices in which $\mathbf{U}{r}=\left[\mathbf{u}{1}, \ldots, \mathbf{u}{r}\right]$ (consisting of $r$ left singular vectors) is an $m \times r$ semi-unitary matrix (cf. (1.75)) due to $$\mathbf{U}{r}^{T} \mathbf{U}{r}=\mathbf{I}{r}$$
and $\mathbf{V}{r}=\left[\mathbf{v}{1}, \ldots, \mathbf{v}{r}\right]$ (consisting of $r$ right singular vectors) is an $n \times r$ semiunitary matrix (i.e., $\mathbf{V}{r}^{T} \mathbf{V}{r}=\mathbf{I}{r}$ ), and
$$\boldsymbol{\Sigma}=\left[\begin{array}{c|c} \operatorname{Diag}\left(\sigma_{1}, \ldots, \sigma_{r}\right) & 0_{r \times(n-r)} \ \hline 0_{(m-r) \times r} & 0_{(m-r) \times(n-r)} \end{array} \in \mathbb{R}^{m \times n}\right.$$
is a rectangular matrix with $r$ positive singular values (supposedly arranged in nonincreasing order), denoted as $\sigma_{i}$, as the first $r$ diagonal elements and zeros elsewhere. Moreover, the range space of $\mathbf{A}$ and that of $\mathbf{U}{r}$ are identical, i.e., $$\mathcal{R}(\mathbf{A})=\mathcal{R}\left(\mathbf{U}{r}\right)$$
The thin $S V D$ of an $m \times n$ matrix $\mathbf{A}$ with rank $r$ is given by
$$\mathbf{A}=\mathbf{U}{r} \boldsymbol{\Sigma}{r} \mathbf{V}{r}^{T}=\sum{i=1}^{r} \sigma_{i} \mathbf{u}{i} \mathbf{v}{i}^{T}$$

## 数学代写|凸优化作业代写Convex Optimization代考|Least-squares approximation

The method of least squares is extensively used to approximately solve for the unknown variables of a linear system with the given set of noisy measurements. Least squares can be interpreted as a method of data fitting. The best fit, between modeled and observed data, in the LS sense is that the sum of squared residuals reaches the least value, where a residual is the difference between an observed value and the value computed from the model.
Consider a system characterized by a set of linear equations,
$$\mathbf{b}=\mathbf{A x}+\boldsymbol{\epsilon},$$
where $\mathbf{A} \in \mathbb{R}^{m \times n}$ is the given system matrix, b is the given data vector, and $\epsilon \in$ $\mathbb{R}^{m}$ is the measurement noise vector. The LS problem is to find an optimal $\mathbf{x} \in \mathbb{R}^{n}$ by minimizing $|\mathbf{A x}-\mathbf{b}|_{2}^{2}$. The LS solution, denoted as $\mathbf{x}{\mathrm{LS}}$, that minimizes $|\mathbf{A x}-\mathbf{b}|{2}^{2}$, is known as
$$\mathbf{x}{\mathrm{LS}} \triangleq \arg \min {\mathbf{x} \in \mathbb{R}^{n}}\left{|\mathbf{A} \mathbf{x}-\mathbf{b}|_{2}^{2}\right}=\mathbf{A}^{\dagger} \mathbf{b}+\mathbf{v}, \mathbf{v} \in \mathcal{N}(\mathbf{A})$$
which is actually an unconstrained optimization problem and the solution (with the same form as the solution (1.119) to the linear equations (1.118)) may not be unique.

As $m \geq n$, the system (1.123) is an over-determined system (i.e., more equations than unknowns), otherwise an under-determined system (i.e., more unknowns than equations). Suppose that $\mathbf{A}$ is of full column rank for the overdetermined case $(m \geq n)$, and thus
$$\mathbf{A}^{\dagger}=\left(\mathbf{A}^{T} \mathbf{A}\right)^{-1} \mathbf{A}^{T}, \quad \mathbf{A}^{\dagger} \mathbf{A}=\mathbf{I}{n}$$ Then the optimal $\mathbf{x}{\mathrm{LS}}=\mathbf{A}^{\dagger} \mathbf{b}$ is unique with the approximation error
$$\left|\mathbf{A} \mathbf{x}{\mathrm{LS}}-\mathbf{b}\right|{2}^{2}=\left|\mathbf{P}{\mathbf{A}}^{\perp} \mathbf{b}\right|{2}^{2}>0 \text {, if } \mathbf{b} \notin \mathcal{R}(\mathbf{A}) \quad \text { (by (1.125) and (1.74)). }$$
In other words, $\mathbf{A} \mathbf{x}_{\mathrm{LS}}$ is the image vector of $\mathbf{b}$ projected on the range space $\mathcal{R}(\mathbf{A})$, and so the approximation error is equal to zero only if $\mathbf{b} \in \mathcal{R}(\mathbf{A})$.

## 数学代写|凸优化作业代写Convex Optimization代考| Singular value decomposition

$$\mathbf{A}=\mathbf{U} \Sigma \mathbf{V}^{T}$$

$$\mathbf{U} r^{T} \mathbf{U} r=\mathbf{I} r$$

$$\mathcal{R}(\mathbf{A})=\mathcal{R}(\mathbf{U} r)$$

$$\mathbf{A}=\mathbf{U} r \boldsymbol{\Sigma} r \mathbf{V} r^{T}=\sum i=1^{r} \sigma_{i} \mathbf{u} i \mathbf{v} i^{T}$$

## 数学代写|凸优化作业代写Convex Optimization代考| Least-squares approximation

$$\mathbf{b}=\mathbf{A} \mathbf{x}+\boldsymbol{\epsilon}$$

$$\mathbf{A}^{\dagger}=\left(\mathbf{A}^{T} \mathbf{A}\right)^{-1} \mathbf{A}^{T}, \quad \mathbf{A}^{\dagger} \mathbf{A}=\mathbf{I} n$$

$$|\mathbf{A x L S}-\mathbf{b}| 2^{2}=\left|\mathbf{P A}^{\perp} \mathbf{b}\right| 2^{2}>0, \text { if } \mathbf{b} \notin \mathcal{R}(\mathbf{A}) \quad \text { (by (1.125) and (1.74)). }$$

## 有限元方法代写

assignmentutor™作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师
The provided text passage discusses the Singular Value Decomposition (SVD) and the method of least squares in the context of linear algebra and matrix operations. It explains the structure of the SVD, the properties of the matrices involved, and the application of the least-squares method to solve over-determined and under-determined systems of linear equations.
no_question

the passage does not contain any explicit mathematical questions or problems to solve. it is purely expository, providing definitions and explanations of the svd and least-squares approximation methods.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
assignmentutor™您的专属作业导师

assignmentutor-lab™ 为您的留学生涯保驾护航 在代写凸优化Convex Optimization方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写凸优化Convex Optimization代写方面经验极为丰富，各种代写凸优化Convex Optimization相关的作业也就用不着说。

• 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|凸优化作业代写Convex Optimization代考|Singular value decomposition

The singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with many applications in signal processing and communications. Applications which employ the SVD include computation of pseudo-inverse, least-squares fitting of data, matrix approximation, and determination of rank, range space, and null space of a matrix and so on and so forth.
Let $\mathbf{A} \in \mathbb{R}^{m \times n}$ with $\operatorname{rank}(\mathbf{A})=r$. The SVD of $\mathbf{A}$ is expressed in the form
$$\mathbf{A}=\mathbf{U} \Sigma \mathbf{V}^{T} \text {. }$$
In the full SVD (1.107) for $\mathbf{A}, \mathbf{U}=\left[\mathbf{U}{r}, \mathbf{U}^{\prime}\right] \in \mathbb{R}^{m \times m}$ and $\mathbf{V}=\left[\mathbf{V}{r}, \mathbf{V}^{\prime}\right] \in \mathbb{R}^{n \times n}$ are orthogonal matrices in which $\mathbf{U}{r}=\left[\mathbf{u}{1}, \ldots, \mathbf{u}{r}\right]$ (consisting of $r$ left singular vectors) is an $m \times r$ semi-unitary matrix (cf. (1.75)) due to $$\mathbf{U}{r}^{T} \mathbf{U}{r}=\mathbf{I}{r}$$
and $\mathbf{V}{r}=\left[\mathbf{v}{1}, \ldots, \mathbf{v}{r}\right]$ (consisting of $r$ right singular vectors) is an $n \times r$ semiunitary matrix (i.e., $\mathbf{V}{r}^{T} \mathbf{V}{r}=\mathbf{I}{r}$ ), and
$$\boldsymbol{\Sigma}=\left[\begin{array}{c|c} \operatorname{Diag}\left(\sigma_{1}, \ldots, \sigma_{r}\right) & 0_{r \times(n-r)} \ \hline 0_{(m-r) \times r} & 0_{(m-r) \times(n-r)} \end{array} \in \mathbb{R}^{m \times n}\right.$$
is a rectangular matrix with $r$ positive singular values (supposedly arranged in nonincreasing order), denoted as $\sigma_{i}$, as the first $r$ diagonal elements and zeros elsewhere. Moreover, the range space of $\mathbf{A}$ and that of $\mathbf{U}{r}$ are identical, i.e., $$\mathcal{R}(\mathbf{A})=\mathcal{R}\left(\mathbf{U}{r}\right)$$
The thin $S V D$ of an $m \times n$ matrix $\mathbf{A}$ with rank $r$ is given by
$$\mathbf{A}=\mathbf{U}{r} \boldsymbol{\Sigma}{r} \mathbf{V}{r}^{T}=\sum{i=1}^{r} \sigma_{i} \mathbf{u}{i} \mathbf{v}{i}^{T}$$

## 数学代写|凸优化作业代写Convex Optimization代考|Least-squares approximation

The method of least squares is extensively used to approximately solve for the unknown variables of a linear system with the given set of noisy measurements. Least squares can be interpreted as a method of data fitting. The best fit, between modeled and observed data, in the LS sense is that the sum of squared residuals reaches the least value, where a residual is the difference between an observed value and the value computed from the model.
Consider a system characterized by a set of linear equations,
$$\mathbf{b}=\mathbf{A x}+\boldsymbol{\epsilon},$$
where $\mathbf{A} \in \mathbb{R}^{m \times n}$ is the given system matrix, b is the given data vector, and $\epsilon \in$ $\mathbb{R}^{m}$ is the measurement noise vector. The LS problem is to find an optimal $\mathbf{x} \in \mathbb{R}^{n}$ by minimizing $|\mathbf{A x}-\mathbf{b}|_{2}^{2}$. The LS solution, denoted as $\mathbf{x}{\mathrm{LS}}$, that minimizes $|\mathbf{A x}-\mathbf{b}|{2}^{2}$, is known as
$$\mathbf{x}{\mathrm{LS}} \triangleq \arg \min {\mathbf{x} \in \mathbb{R}^{n}}\left{|\mathbf{A} \mathbf{x}-\mathbf{b}|_{2}^{2}\right}=\mathbf{A}^{\dagger} \mathbf{b}+\mathbf{v}, \mathbf{v} \in \mathcal{N}(\mathbf{A})$$
which is actually an unconstrained optimization problem and the solution (with the same form as the solution (1.119) to the linear equations (1.118)) may not be unique.

As $m \geq n$, the system (1.123) is an over-determined system (i.e., more equations than unknowns), otherwise an under-determined system (i.e., more unknowns than equations). Suppose that $\mathbf{A}$ is of full column rank for the overdetermined case $(m \geq n)$, and thus
$$\mathbf{A}^{\dagger}=\left(\mathbf{A}^{T} \mathbf{A}\right)^{-1} \mathbf{A}^{T}, \quad \mathbf{A}^{\dagger} \mathbf{A}=\mathbf{I}{n}$$ Then the optimal $\mathbf{x}{\mathrm{LS}}=\mathbf{A}^{\dagger} \mathbf{b}$ is unique with the approximation error
$$\left|\mathbf{A} \mathbf{x}{\mathrm{LS}}-\mathbf{b}\right|{2}^{2}=\left|\mathbf{P}{\mathbf{A}}^{\perp} \mathbf{b}\right|{2}^{2}>0 \text {, if } \mathbf{b} \notin \mathcal{R}(\mathbf{A}) \quad \text { (by (1.125) and (1.74)). }$$
In other words, $\mathbf{A} \mathbf{x}_{\mathrm{LS}}$ is the image vector of $\mathbf{b}$ projected on the range space $\mathcal{R}(\mathbf{A})$, and so the approximation error is equal to zero only if $\mathbf{b} \in \mathcal{R}(\mathbf{A})$.

## 数学代写|凸优化作业代写Convex Optimization代考| Singular value decomposition

$$\mathbf{A}=\mathbf{U} \Sigma \mathbf{V}^{T}$$

$$\mathbf{U} r^{T} \mathbf{U} r=\mathbf{I} r$$

$$\mathcal{R}(\mathbf{A})=\mathcal{R}(\mathbf{U} r)$$

$$\mathbf{A}=\mathbf{U} r \boldsymbol{\Sigma} r \mathbf{V} r^{T}=\sum i=1^{r} \sigma_{i} \mathbf{u} i \mathbf{v} i^{T}$$

## 数学代写|凸优化作业代写Convex Optimization代考| Least-squares approximation

$$\mathbf{b}=\mathbf{A} \mathbf{x}+\boldsymbol{\epsilon}$$

$$\mathbf{A}^{\dagger}=\left(\mathbf{A}^{T} \mathbf{A}\right)^{-1} \mathbf{A}^{T}, \quad \mathbf{A}^{\dagger} \mathbf{A}=\mathbf{I} n$$

$$|\mathbf{A x L S}-\mathbf{b}| 2^{2}=\left|\mathbf{P A}^{\perp} \mathbf{b}\right| 2^{2}>0, \text { if } \mathbf{b} \notin \mathcal{R}(\mathbf{A}) \quad \text { (by (1.125) and (1.74)). }$$

## 有限元方法代写

assignmentutor™作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师
user
The provided text passage discusses the Singular Value Decomposition (SVD) and the method of least squares in the context of linear algebra and matrix operations. It explains the structure of the SVD, the properties of the matrices involved, and the application of the least-squares method to solve over-determined and under-determined systems of linear equations.

TYPE: NO_QUESTION

The passage does not contain any explicit mathematical questions or problems to solve. It is purely expository, providing definitions and explanations of the SVD and least-squares approximation methods.