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

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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|Properties of SVD

Assume $\boldsymbol{A} \in \Re^{m \times n}, \quad \boldsymbol{B} \in \Re^{m \times n}$, and $r_{A}=\operatorname{rank}(A), \quad p=\min {m, n}$. The singular values of matrix $A$ can be arranged as follows: $\sigma_{\max }=\sigma_{1} \geq \sigma_{2} \geq \cdots$ $\geq \sigma_{p-1} \geq \sigma_{p}=\sigma_{\min } \geq 0$, and denote by $\sigma_{i}(\boldsymbol{B})$ the $i$ th largest singular value of matrix B. A few properties of SVD can summarized as follows [6]:
(1) The relationship between the singular values of a matrix and the ones of its submatrix.

Theorem $2.3$ (interlacing theorem for singular values). Assume $A \in \Re^{m \times n}$, and its singular values satisfy $\sigma_{1} \geq \sigma_{2} \geq \cdots \geq \sigma_{r}$, where $r=\min {m, n} .$ If $\boldsymbol{B} \in \mathbb{X}^{p \times q}$ is a submatrix of $\boldsymbol{A}$, and its singular values satisfy $\gamma_{1} \geq \gamma_{2} \geq \cdots \geq \gamma_{\min {p, q}}$, then it holds that
$$\sigma_{i} \geq \gamma_{i}, \quad i=1,2, \ldots, \min {p, q}$$
and
$$\gamma_{i} \geq \sigma_{i+(m-p)+(n-q)}, \quad i \leq \min {p+q-m, p+q-n} .$$
From Theorem 2.3, it holds that: If $\boldsymbol{B} \in \Re^{m \times(n-1)}$ is a submatrix of $\mathbf{A} \in \Re^{m \times n}$ by deleting any column of matrix $\boldsymbol{A}$, and their singular values are arranged in non-decreasing order, then it holds that
$$\sigma_{1}(\boldsymbol{A}) \geq \sigma_{1}(\boldsymbol{B}) \geq \sigma_{2}(\boldsymbol{A}) \geq \sigma_{2}(\boldsymbol{B}) \geq \cdots \geq \sigma_{h}(\boldsymbol{A}) \geq \sigma_{h}(\boldsymbol{B}) \geq 0$$
where $h=\min {m, n-1}$.
If $\boldsymbol{B} \in \Re^{\Re^{(m-1) \times n}}$ is a submatrix of $\boldsymbol{A} \in \Re^{m \times n}$ by deleting any row of matrix $\boldsymbol{A}$, and their singular values are arranged as non-decreasing order, then it holds that
$$\sigma_{1}(\boldsymbol{A}) \geq \sigma_{1}(\boldsymbol{B}) \geq \sigma_{2}(\boldsymbol{A}) \geq \sigma_{2}(\boldsymbol{B}) \geq \cdots \sigma_{h}(\boldsymbol{A}) \geq \sigma_{h}(\boldsymbol{B}) \geq 0$$
(2) The relationship between the singular values of a matrix and its norms. The spectral norm of a matrix $\boldsymbol{A}$ is equal to its largest singular value, namely,

According to the SVD theorem of matrix and the unitary invariability property of Frobenius norm $|\boldsymbol{A}|_{F}$ of matrix $\boldsymbol{A}$, namely $\left|\boldsymbol{U}^{H} \boldsymbol{A} \boldsymbol{V}\right|_{F}=|\boldsymbol{A}|_{F}$, it holds that
$$|\boldsymbol{A}|_{F}=\left[\sum_{i=1}^{m} \sum_{j=1}^{n}\left|a_{i j}\right|^{2}\right]^{1 / 2}=\left|\boldsymbol{U}^{H} \boldsymbol{A} \boldsymbol{V}\right|_{F}=|\Sigma|_{F}=\sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}+\cdots+\sigma_{F}^{2}}$$
That is to say, the Frobenius norm of any matrix is equal to the square root of the sum of the squares of all nonzero singular values of this matrix. Consider the rank- $k$ approximation of matrix $A$ and denote it as $\boldsymbol{A}{k}$, in which $k{k}$ is defined as follows:
$$A_{k}=\sum_{i=1}^{k} \sigma_{i} \boldsymbol{u}{i} v{i}^{H}, k<r$$

## 机器学习代写|主成分分析作业代写PCA代考|Eigenvalue Problem and Eigen Equation

The basic problem of the eigenvalue can be stated as follows. Given an $n \times n$ matrix $\boldsymbol{A}$, determine a scalar $\lambda$ such that the following algebra equation
$$A u=\lambda u, \quad u \neq 0$$
has an $n \times 1$ nonzero solution. The scalar $\lambda$ is called as an eigenvalue of matrix $A$, and the vector $\boldsymbol{u}$ is called as the eigenvector associated with $\lambda$. Since the eigenvalue
$\lambda$ and eigenvector $\boldsymbol{u}$ appear in couples, $(\lambda, \boldsymbol{u})$ is usually called as an eigen pair of matrix $\boldsymbol{A}$. Although the eigenvalues can be zeros, the eigenvectors cannot be zero. In order to determine a nonzero vector $\boldsymbol{u}, \mathrm{Eq} .$ (2.17) can be modified as
$$(A-\lambda I) u=0$$
The above equation should come into existence for any vector $\boldsymbol{u}$, so the unique condition under which Eq. $(2.18)$ has a nonzero solution $\boldsymbol{u}=0$ is that the determinant of matrix $\boldsymbol{A}-\lambda \boldsymbol{I}$ is equal to zero, namely
$$\operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})=0 .$$
Thus, the solution of the eigenvalue problem consists of the following two steps:
(1) Solve all scalar $\lambda$ (eigenvalues) which make the matrix $\boldsymbol{A}-\lambda \boldsymbol{I}$ singular.
(2) Given an eigenvalue $\lambda$ which makes $\boldsymbol{A}=\lambda \boldsymbol{I}$ singular, and to solve all nonzero vectors which meets $(\boldsymbol{A}-\lambda \boldsymbol{I}) \boldsymbol{x}=\boldsymbol{0}$, i.e., the eigenvectors corresponding to $\lambda$.
According to the relationship between the singular values of a matrix and its determinant, a matrix is singular if and only if $\operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})=\boldsymbol{0}{,}$, namely $$(\boldsymbol{A}-\lambda \boldsymbol{I}) x \text { singular } \Leftrightarrow \operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})=\boldsymbol{0}$$ The matrix $(\boldsymbol{A}-\lambda \boldsymbol{I})$ is called as the eigen matrix of $\boldsymbol{A}$. When $\boldsymbol{A}$ is an $n \times n$ matrix, spreading the left side determinant of Eq. (2.20) can obtain a polynomial equation (power- $n$ ), namely $$\alpha{0}+\alpha_{1} \lambda+\cdots+\alpha_{n-1} \lambda^{n-1}+(-1)^{n} \lambda^{n}=0,$$
which is called as the eigen equation of matrix $\boldsymbol{A}$. The polynomial $\operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})$ is called as the eigen polynomial.

## 机器学习代写|主成分分析作业代写PCA代考|Eigenvalue and Eigenvector

In the following, we list some major properties about the eigenvalues and eigenvector of a matrix $A$.
Several important terms about the eigenvalues and eigenvectors [6]:
(1) The eigenvalue $\lambda$ of a matrix $A$ is called as having algebraic multiplicity $\mu$, if $\lambda$ is a $\mu$-repeated root of the eigen equation $\operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{I})=\boldsymbol{0}$.
(2) If the algebraic multiplicity of eigenvalue $\lambda$ is equal to one, the eigenvalue is called as single eigenvalue. Non-single eigenvalues are called as multiple eigenvalues.
(3) The eigenvalue $\lambda$ of a matrix $\boldsymbol{A}$ is called as having geometric multiplicity $\gamma$, if the number of linear independent eigenvectors associated with $\lambda$ is equal to $\gamma$.
(4) An eigenvalue is called half-single eigenvalue if its algebraic multiplicity is equal to geometric multiplicity. Not half-single eigenvalues are called as wane eigenvalues.
(5) If matrix $\boldsymbol{A}{n \times n}$ is a general complex matrix and $\lambda$ is its eigenvalue, the vector $v$ which meets $A v=\lambda v$ is called as the right eigenvector associated with the eigenvalue $\lambda$, and the eigenvector $\boldsymbol{u}$ which meets $\boldsymbol{u}^{H} \boldsymbol{A}=\lambda \boldsymbol{u}^{H}$ is called as the left eigenvector associated with the eigenvalue $\lambda$. If $A$ is Hermitian matrix and all its eigenvalues are real number, then it holds that $\boldsymbol{v}=\boldsymbol{u}$, that is to say, the left and right eigenvectors of a Hermitian matrix are the same. Some important properties can be summarized as follows: (1) Matrix $A\left(\in \Re^{n \times n}\right)$ has $n$ eigenvalues, of which the multiple eigenvalues are computed according to their multiplicity. (2) If $\boldsymbol{A}$ is a real symmetrical matrix or Hermitian matrix, all its eigenvalues are real numbers. (3) If $\boldsymbol{A}=\operatorname{diag}\left(a{11}, a_{22}, \ldots, a_{\mathrm{nn}}\right)$, its eigenvalues are $a_{11}, a_{22}, \ldots, a_{\mathrm{nn}}$; If $\boldsymbol{A}$ is a trigonal matrix, its diagonal elements are all its eigenvalues.
(4) For $\boldsymbol{A}\left(\in \Re^{n \times n}\right)$, if $\lambda$ is the eigenvalue of matrix $\boldsymbol{A}, \lambda$ is also the eigenvalue of matrix $A^{\mathrm{T}}$. If $\lambda$ is the eigenvalue of matrix $A, \lambda^{*}$ is the eigenvalue of matrix $A^{H}$. If $\lambda$ is the eigenvalue of matrix $A, \lambda+\sigma^{2}$ is the eigenvalue of matrix $\boldsymbol{A}+\sigma^{2} \boldsymbol{I}$. If $\lambda$ is the eigenvalue of matrix $\boldsymbol{A}, 1 / \lambda$ is the eigenvalue of matrix $A^{-1}$.
(5) All eigenvalues of matrix $A^{2}=A$ are either 0 or 1 .
(6) If $A$ is a real orthogonal matrix, all its eigenvalues are on the unit circle.
(7) If a matrix is singular, at least one of its eigenvalues is equal to zero.
(8) The sum of all the eigenvalues is equal to its trace, namely $\sum_{i=1}^{n} \lambda_{i}=\operatorname{tr}(\boldsymbol{A})$.
(9) The nonzero eigenvectors $\boldsymbol{u}{1}, \boldsymbol{u}{2}, \ldots, \boldsymbol{u}{n}$ associated with different eigenvalues $\lambda{1}, \lambda_{2}, \ldots \lambda_{n}$ are linearly independent.
(10) If matrix $A\left(\in \mathcal{H}^{\mathrm{n} \times n}\right)$ has $r$ nonzero eigenvalues, then it holds that $\operatorname{rank}(\boldsymbol{A}) \geq r$; If zero is a non-multiple eigenvalue, then $\operatorname{rank}(\boldsymbol{A}) \geq n-1$; If $\operatorname{rank}(\boldsymbol{A}-\lambda \boldsymbol{I}) \geq n-1$, then $\lambda$ is an eigenvalue of matrix $\boldsymbol{A}$.
(11) The product of all eigenvalues of matrix $A$ is equal to the determinant of matrix $\boldsymbol{A}$, namely $\prod_{i=1}^{n} \lambda_{i}=\operatorname{det}(\boldsymbol{A})=|\boldsymbol{A}|$.
(12) A Hermitian matrix $\boldsymbol{A}$ is positive definite (or positive semi-definite), if and only if all its eigenvalues are positive (or non-negative).

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

(1)矩阵的奇异值与其子矩阵的奇异值之间的关系。

σ一世≥C一世,一世=1,2,…,分钟p,q

C一世≥σ一世+(米−p)+(n−q),一世≤分钟p+q−米,p+q−n.

σ1(一种)≥σ1(乙)≥σ2(一种)≥σ2(乙)≥⋯≥σH(一种)≥σH(乙)≥0

σ1(一种)≥σ1(乙)≥σ2(一种)≥σ2(乙)≥⋯σH(一种)≥σH(乙)≥0
(2) 矩阵的奇异值与其范数之间的关系。矩阵的谱范数一种等于它的最大奇异值，即

|一种|F=[∑一世=1米∑j=1n|一种一世j|2]1/2=|在H一种在|F=|Σ|F=σ12+σ22+⋯+σF2

## 机器学习代写|主成分分析作业代写PCA代考|Eigenvalue Problem and Eigen Equation

λ和特征向量在出现在情侣中，(λ,在)通常称为矩阵的特征对一种. 虽然特征值可以为零，但特征向量不能为零。为了确定一个非零向量在,和q.(2.17) 可以修改为
(一种−λ一世)在=0

(1) 求解所有标量λ（特征值）构成矩阵一种−λ一世单数。
(2) 给定一个特征值λ这使得一种=λ一世奇异的，并求解所有满足的非零向量(一种−λ一世)X=0，即对应的特征向量λ.

## 机器学习代写|主成分分析作业代写PCA代考|Eigenvalue and Eigenvector

1）特征值λ矩阵的一种被称为具有代数多重性μ， 如果λ是一个μ-特征方程的重复根这⁡(一种−λ一世)=0.
(2) 若特征值的代数重数λ等于一，则该特征值称为单一特征值。非单一特征值称为多重特征值。
(3) 特征值λ矩阵的一种被称为具有几何多重性C，如果与相关的线性独立特征向量的数量λ等于C.
(4) 如果一个特征值的代数重数等于几何重数，则称其为半单特征值。非半单特征值称为衰减特征值。
(5) 如果矩阵一种n×n是一个一般的复矩阵，并且λ是它的特征值，向量在满足一种在=λ在被称为与特征值相关的右特征向量λ, 和特征向量在满足在H一种=λ在H被称为与特征值相关的左特征向量λ. 如果一种是 Hermitian 矩阵并且它的所有特征值都是实数，那么它认为在=在，也就是说，一个厄密矩阵的左右特征向量是相同的。一些重要的性质可以概括如下： (1) 矩阵一种(∈ℜn×n)拥有n特征值，其中的多个特征值是根据它们的多重性计算的。(2) 如果一种是一个实对称矩阵或 Hermitian 矩阵，它的所有特征值都是实数。(3) 如果一种=诊断⁡(一种11,一种22,…,一种nn)，其特征值为一种11,一种22,…,一种nn; 如果一种是一个三角矩阵，它的对角元素都是它的特征值。
(4) 为一种(∈ℜn×n)， 如果λ是矩阵的特征值一种,λ也是矩阵的特征值一种吨. 如果λ是矩阵的特征值一种,λ∗是矩阵的特征值一种H. 如果λ是矩阵的特征值一种,λ+σ2是矩阵的特征值一种+σ2一世. 如果λ是矩阵的特征值一种,1/λ是矩阵的特征值一种−1.
(5) 矩阵的所有特征值一种2=一种是 0 或 1 。
(6) 如果一种是一个实正交矩阵，它的所有特征值都在单位圆上。
(7) 如果一个矩阵是奇异的，至少它的一个特征值等于 0。
(8) 所有特征值之和等于它的迹，即∑一世=1nλ一世=tr⁡(一种).
(9) 非零特征向量在1,在2,…,在n与不同的特征值相关联λ1,λ2,…λn是线性独立的。
(10) 如果矩阵一种(∈Hn×n)拥有r非零特征值，那么它认为秩⁡(一种)≥r; 如果零是非多重特征值，则秩⁡(一种)≥n−1; 如果秩⁡(一种−λ一世)≥n−1， 然后λ是矩阵的特征值一种.
(11) 矩阵所有特征值的乘积一种等于矩阵的行列式一种，即∏一世=1nλ一世=这⁡(一种)=|一种|.
(12) Hermitian 矩阵一种是正定的（或半正定的），当且仅当它的所有特征值都是正的（或非负的）。

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## MATLAB代写

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