### 机器学习代写|主成分分析作业代写PCA代考| Matrix Analysis Basics

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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

## 机器学习代写|主成分分析作业代写PCA代考|Matrix Analysis Basics

As discussed in Chap. 1, the $\mathrm{PC}$ or $\mathrm{MC}$ can be obtained by the ED of the sample correlation matrix or the SVD of the data matrix, and ED and SVD are also primal analysis tools. The history of SVD can date back to the $1870 \mathrm{~s}$, and Beltrami and Jordan are acknowledged as the founder of SVD. In 1873, Beltrami [1] published the first paper on SVD, and one year later Jordan [2] published his independent reasoning about SVD. Now, SVD has become one of the most useful and most efficient modern numerical analysis tools, and it has been widely used in statistical analysis, signal and image processing, system theory and control, etc. SVD is also a fundamental tool for eigenvector extraction, subspace tracking, and total least squares problem, etc.

On the other hand, ED is important in both mathematical analysis and engineering applications. For example, in matrix algebra, ED is usually related to the spectral analysis, and the spectral of a linear arithmetic operator is defined as the set of eigenvalues of the matrix. In engineering applications, spectral analysis is connected to the Fourier analysis, and the frequency spectral of signals is defined as the Fourier spectral, and then the power spectral of signals is defined as the square of frequency spectral norm or Fourier transform of the autocorrelation functions.
Besides SVD and ED, gradient and matrix differential are also the important concepts of matrix analysis. In view of the use of them in latter chapters, we will provide detailed analysis of SVD, ED, matrix analysis, etc. in the following.

## 机器学习代写|主成分分析作业代写PCA代考|Singular Value Decomposition

As to the inventor history of SVD, see Stewart’s dissertation. Later, Autonne [3] extended SVD to complex square matrix in 1902, and Eckart and Young [4] further extended it to general rectangle matrix in 1939. Now, the theorem of SVD for rectangle matrix is usually called Eckart-Young Theorem.

SVD can be viewed as the extension of ED to the case of nonsquare matrices. It says that any real matrix can be diagonalized by using two orthogonal matrices. ED works only for square matrices and uses only one matrix (and its inverse) to achieve diagonalization. If the matrix is square and symmetric, then the two orthogonal matrices of SVD will be the same, and ED and SVD will also be the same and closely related to the matrix rank and reduced-rank least squares approximations.

## 机器学习代写|主成分分析作业代写PCA代考|Theorem and Uniqueness of SVD

Theorem 2.1 For any $\mathbf{A} \in \Re^{m \times n}$ (or $\mathbb{C}^{m \times n}$ ), there exist two orthonormal (or unitary) matrices $U \in \Re^{m \times n}$ (or $\mathbb{C}^{m \times m}$ ) and $\mathbf{V} \in \mathfrak{R}^{m \times n}$ (or $\mathbb{C}^{n \times n}$ ), such that
$$\boldsymbol{A}=\boldsymbol{U} \Sigma V^{\mathrm{T}}\left(\text { or } A=\boldsymbol{U} \Sigma V^{H}\right),$$
where,
$$\Sigma=\left[\begin{array}{cc} \Sigma_{1} & 0 \ 0 & 0 \end{array}\right]$$
and $\boldsymbol{\Sigma}=\operatorname{diag}\left[\sigma_{1}, \sigma_{2}, \ldots \sigma_{r}\right]$, its diagonal elements are arranged in the order:
$$\sigma_{1} \geq \sigma_{2} \geq \cdots \geq \sigma_{r} \geq 0, \quad t=\operatorname{rank}(A)$$
The quantity $\sigma_{1}, \sigma_{2}, \ldots, \sigma_{r}$ together with $\sigma_{r+1}=\sigma_{r+2}=\cdots=\sigma_{n}=0$ are called the singular values of matrix $\boldsymbol{A}$. The column vector $\boldsymbol{u}{i}$ of matrix $\boldsymbol{U}$ is called the left singular vector of $\boldsymbol{A}$, and the matrix $\boldsymbol{U}$ is called the left singular matrix. The column vector $v{i}$ of matrix $\boldsymbol{V}$ is called the right singular vector of $\boldsymbol{A}$, and the matrix $V$ is called the right singular matrix. The proof of Theorem $2.1$ can see $[4,5]$. The SVD of matrix $A$ can also be written as:
$$\boldsymbol{A}=\sum_{i=1}^{r} \sigma_{i} \boldsymbol{u}{i} v{i}^{H}$$

$$\boldsymbol{A} \boldsymbol{A}^{H}=\boldsymbol{U} \Sigma^{2} \boldsymbol{U}^{H}$$
which shows that the singular value $\sigma_{i}$ of the $m \times n$ matrix $\boldsymbol{A}$ is the positive square root of the eigenvalue (these eigenvalues are nonpositive) of the matrix product $A A^{\mathrm{H}}$.
The following theorem strictly narrates the singular property of a matrix $A$.
Theorem 2.2 Define the singular values of matrix $\mathbf{A} \in \AA^{\mathrm{m} \times n}(m>n)$ as $\sigma_{1} \geq \sigma_{2} \geq \cdots \geq \sigma_{r} \geq 0 .$
Then
$$\sigma_{k}=\min {E \in \mathbb{C}^{m x}}\left{|\boldsymbol{E}|{s p e c}: \operatorname{rank}(\boldsymbol{A}+\boldsymbol{E}) \leq(k-1)\right}, \quad k=1,2, \ldots n$$
and there is an error matrix which meets $\left|\boldsymbol{E}{k}\right|{\text {spec }}=\sigma_{k}$, so that
$$\operatorname{rank}\left(\boldsymbol{A}+\boldsymbol{E}{k}\right)=r-1, \quad k=1,2, \ldots, n .$$ Theorem $2.2$ shows that the singular value of a matrix is equal to the spectral norm of the error matrix $\boldsymbol{E}{k}$ which makes the rank of the original matrix reduce one. If the original $n \times n$ matrix $\boldsymbol{A}$ is square and it has a zero singular value, the spectral norm of error matrix whose rank reduces to one is equal to zero. That is to say, when the original $n \times n$ matrix $\boldsymbol{A}$ has a zero singular value, the rank of the matrix is $\operatorname{rank}(\mathbf{A}) \leq n-1$ and the original matrix is not full-rank essentially. So, if a matrix has a zero singular value, the matrix must be singular matrix. Generally speaking, if a rectangle matrix has a zero singular value, then it must not be full column rank or full row rank. This case is called rank-deficient matrix, which is a singular phenomenon with regards to the full-rank matrix.

## 机器学习代写|主成分分析作业代写PCA代考|Singular Value Decomposition

SVD 可以看作是 ED 对非方阵情况的扩展。它说任何实矩阵都可以通过使用两个正交矩阵进行对角化。ED 仅适用于方阵并且仅使用一个矩阵（及其逆矩阵）来实现对角化。如果矩阵是正方形且对称的，那么 SVD 的两个正交矩阵将是相同的，ED 和 SVD 也将是相同的，并且与矩阵秩和降秩最小二乘逼近密切相关。

## 机器学习代写|主成分分析作业代写PCA代考|Theorem and Uniqueness of SVD

Σ=[Σ10 00]

σ1≥σ2≥⋯≥σr≥0,吨=秩⁡(一种)

\sigma_{k}=\min {E \in \mathbb{C}^{m x}}\left{|\boldsymbol{E}|{s p e c}: \operatorname{rank}(\boldsymbol{A}+\boldsymbol {E}) \leq(k-1)\right}, \quad k=1,2, \ldots n\sigma_{k}=\min {E \in \mathbb{C}^{m x}}\left{|\boldsymbol{E}|{s p e c}: \operatorname{rank}(\boldsymbol{A}+\boldsymbol {E}) \leq(k-1)\right}, \quad k=1,2, \ldots n

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