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

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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|Eigenvalue Decomposition of Hermitian Matrix

All the discussions on eigenvalues and eigenvectors in the above hold for general matrices, and they do not require the matrices to be real symmetric or complex conjugate symmetric. However, in the statistical and information science, one usually encounter real symmetric or Hermitian (complex conjugate symmetric) matrices. For example, the autocorrelation matrix of a real measurement data vector $\boldsymbol{R}=E\left{\boldsymbol{x}(t) \boldsymbol{x}^{T}(t)\right}$ is real symmetric, while the autocorrelation matrix of a complex measurement data vector $\boldsymbol{R}=E\left{\boldsymbol{x}(t) \boldsymbol{x}^{H}(t)\right}$ is Hermitian. On the other hand, since a real symmetric matrix is a special case of Hermitian matrix and the eigenvalues and eigenvectors of a Hermitian matrix have a series of important properties, and it is necessary to discuss individually the eigen analysis of Hermitian matrix.

1. Eigenvalue and Eigenvector of Hermitian matrix.
Some important properties of eigenvalues and eigenvectors of Hermitian matrices can be summarized as follows:
(1) The eigenvalues of an Hermitian matrix $A$ must be a real number.
(2) Let $(\lambda, \boldsymbol{u})$ be an eigen pair of an Hermitian matrix $\boldsymbol{A}$. If $\boldsymbol{A}$ is invertible, then $(1 / \lambda, u)$ is an eigen pair of matrix $A^{-1}$.
(3) If $\lambda_{k}$ is a multiple eigenvalue of Hermitian matrix $A^{H}=A$, and its multiplicity is $m_{k}$, then $\operatorname{rank}\left(\boldsymbol{A}-\lambda_{k} \boldsymbol{I}\right)=n-m_{k}$.
(4) Any Hermitian matrix $A$ is diagonalizable, namely $U^{-1} \boldsymbol{A} U=\Sigma$.
(5) All the eigenvectors of an Hermitian matrix are linearly independent, and they are mutual orthogonal, namely the eigen matrix $\boldsymbol{U}=\left[\boldsymbol{u}{1}, \boldsymbol{u}{2}, \ldots, \boldsymbol{u}{n}\right]$ is a unitary matrix and it meets $\boldsymbol{U}^{-1}=\boldsymbol{U}^{H}$. (6) From property (5), it holds that $\boldsymbol{U}^{H} \boldsymbol{A} \boldsymbol{U}=\Sigma=\operatorname{diag}\left(\lambda{1}, \lambda_{2}, \ldots, \lambda_{n}\right)$ or $\boldsymbol{A}=\boldsymbol{U} \Sigma \boldsymbol{U}^{H}$, which can be rewritten as: $\boldsymbol{A}=\sum_{i=1}^{n} \lambda_{i} \boldsymbol{u}{i} \boldsymbol{u}{i}^{H}$. This is called the spectral decomposition of a Hermitian matrix.
(7) The spread formula of the inverse of an Hermitian matrix $A$ is
$$\boldsymbol{A}^{-1}=\sum_{i=1}^{n} \frac{\mathrm{I}}{\lambda_{i}} \boldsymbol{u}{i} \boldsymbol{u}{i}^{H}$$
Thus, if one know the eigen decomposition of an Hermitian matrix $\boldsymbol{A}$, then one can directly obtain the inverse matrix $A^{-1}$ using the above formula.
(8) For two $n \times n$ Hermitian matrices $\boldsymbol{A}$ and $\boldsymbol{B}$, there exists a unitary matrix so that $\boldsymbol{P}^{H} \boldsymbol{A} \boldsymbol{P}$ and $\boldsymbol{P}^{H} \boldsymbol{B P}$ are both diagonal if and only if $\boldsymbol{A B}=\boldsymbol{B A}$.
(9) For two $n \times n$ non-negative definite Hermitian matrices $\boldsymbol{A}$ and $\boldsymbol{B}$, there exists a nonsingular matrix $P$ so that $P^{H} A P$ and $P^{H} B P$ are both diagonal.

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

Let $A$ and $B$ both be $n \times n$ square matrices, and they constitute a matrix pencil or matrix pair, written as $(\boldsymbol{A}, \boldsymbol{B})$. Now we consider the following generalized eigenvalue problem. That is, to compute all scalar $\lambda$ such that
$$A u=\lambda B u$$
has nonzero solution $\boldsymbol{u} \neq 0$, where the scalar $\lambda$ and the nonzero vector $\boldsymbol{u}$ are called the generalized eigenvalue and the generalized eigenvector of matrix pencil $(\boldsymbol{A}, \boldsymbol{B})$, respectively. A generalized eigenvalue and its associated generalized eigenvector are called generalized eigen pair, written as $(\lambda, \boldsymbol{u})$. Equation (2.35) is also called the generalized eigen equation. It is obvious that the eigenvalue problem is a special case when the matrix pencil is chosen as $(\boldsymbol{A}, \boldsymbol{I})$.

Theorem 2.6 $\lambda \in \mathbb{C}$ and $\mathbf{u} \in \mathbb{C}^{n}$ are respectively the generalized eigenvalue and the associated generalized eigenvector of matrix pencil $(\boldsymbol{A}, \boldsymbol{B})_{n \times n}$ if and only if:
(1) $\operatorname{det}(\boldsymbol{A}-\lambda \boldsymbol{B})=0$.
(2) $\boldsymbol{u} \in \operatorname{Null}(\boldsymbol{A}-\lambda \boldsymbol{B})$, and $\boldsymbol{u} \neq 0$.
In the natural science, sometimes it is necessary to discuss the eigenvalue problem of the generalized matrix pencil.

Suppose that $n \times n$ square matrices $\boldsymbol{A}$ and $\boldsymbol{B}$ are both Hermitian, and $B$ is positive definite. Then $(\boldsymbol{A}, \boldsymbol{B})$ is called the regularized matrix pencil.

The eigenvalue problem of regularized matrix pencil is similar to the one of Hermitian matrix.

## 机器学习代写|主成分分析作业代写PCA代考|Rayleigh Quotient

Definition 2.1 The Rayleigh quotient (RQ) of an Hermitian matrix $C \in \mathbb{C}^{n \times n}$ is a scalar, defined as
$$r(\boldsymbol{u})=r(\boldsymbol{u}, \boldsymbol{C})=\frac{\boldsymbol{u}^{H} \boldsymbol{C} \boldsymbol{u}}{\boldsymbol{u}^{H} \boldsymbol{u}},$$
where $u$ is a quantity to be selected. The objective is to maximize or minimize the Rayleigh quotient.
The most relevant properties of the $R Q$ are can be summarized as follows:
(1) Homogeneity: $r(\alpha \boldsymbol{u}, \beta \boldsymbol{u})=\beta r(\boldsymbol{u}, \boldsymbol{C}) \quad \forall \alpha, \beta \neq 0$.
(2) Translation invariance: $\boldsymbol{r}(\boldsymbol{u}, \boldsymbol{C}-\alpha \boldsymbol{I})=\boldsymbol{r}(\boldsymbol{u}, \boldsymbol{C})-\alpha$.
(3) Boundedness: Since $\boldsymbol{u}$ ranges over all nonzero vectors, $r(\boldsymbol{u})$ fills a region in the complex plane which is called the field of values of $\boldsymbol{C}$. This region is closed, bounded, and convex. If $\boldsymbol{C}=\boldsymbol{C}^{*}$ (selfadjoint matrix), the field of values is the real interval bounded by the extreme eigenvalues.
(4) Orthogonality: $\boldsymbol{u} \perp(\boldsymbol{C}-r(\boldsymbol{u}) \boldsymbol{I}) \boldsymbol{u}$.
(5) Minimal residual: $\forall \boldsymbol{u} \neq 0 \wedge \forall$ scalar $\mu,|(\boldsymbol{C}-r(\boldsymbol{u}) \boldsymbol{I}) \boldsymbol{u}| \leq|(\boldsymbol{C}-\mu \boldsymbol{I}) \boldsymbol{u}|$.
Proposition $2.1$ (Stationarity) Let $C$ be a real symmetric n-dimensional matrix with eigenvalues $\lambda_{n} \leq \lambda_{n-1} \leq \cdots \lambda_{1}$ and associated unit eigenvectors $z_{1}, z_{2}, \ldots, z_{n}$. Then it holds that $\lambda_{1}=\max r(\boldsymbol{u}, \boldsymbol{C}), \lambda_{n}=\min r(\boldsymbol{u}, \boldsymbol{C})$. More generally, the critical points and critical values of $r(\boldsymbol{u}, \boldsymbol{C})$ are the eigenvectors and eigenvalues of $\boldsymbol{C}$.

Proposition $2.2$ (Degeneracy): The $R Q$ critical points are degenerate because at these points the Hessian matrix is not invertible. Then the RQ is not a Morse function in every open subspace of the domain containing a critical point.

Furthermore, the following important theorems also holds for RQ.
Courant-Fischer Theorem: Let $C \in \mathbb{C}^{n \times n}$ be an Hermitian matrix, and its eigenvalues are $\lambda_{1} \geq \lambda_{2} \geq \cdots \leq \lambda_{n}$, then it holds that for $\lambda_{k}(1 \leq k \leq u)$ :
$$\lambda_{k}=\min {S, \operatorname{dim}(S)=\boldsymbol{n}-k+1} \max {\boldsymbol{u} \in S, \boldsymbol{u} \neq 0}\left(\frac{\boldsymbol{u}^{H} \boldsymbol{C} \boldsymbol{u}}{\boldsymbol{u}^{H} \boldsymbol{u}}\right)$$
The Courant-Fischer Theorem can also written as
$$\lambda_{k}=\min {S, \operatorname{dim}(S)=k} \max {\boldsymbol{u} \in S, \boldsymbol{u} \neq 0}\left(\frac{\boldsymbol{u}^{H} \boldsymbol{C} \boldsymbol{u}}{\boldsymbol{u}^{H} \boldsymbol{u}}\right)$$

## 机器学习代写|主成分分析作业代写PCA代考|Eigenvalue Decomposition of Hermitian Matrix

1. Hermitian 矩阵的特征值和特征向量。
Hermitian 矩阵的特征值和特征向量的一些重要性质可以概括如下：
(1) Hermitian 矩阵的特征值一种必须是实数。
(2) 让(λ,在)是 Hermitian 矩阵的特征对一种. 如果一种是可逆的，那么(1/λ,在)是矩阵的特征对一种−1.
(3) 如果λķ是 Hermitian 矩阵的多重特征值一种H=一种，其多重性为米ķ， 然后秩⁡(一种−λķ一世)=n−米ķ.
(4) 任何 Hermitian 矩阵一种是可对角化的，即在−1一种在=Σ.
(5) Hermitian矩阵的所有特征向量都是线性独立的，并且相互正交，即特征矩阵在=[在1,在2,…,在n]是酉矩阵并且满足在−1=在H. (6) 根据性质 (5)，它认为在H一种在=Σ=诊断⁡(λ1,λ2,…,λn)或者一种=在Σ在H，可以改写为：一种=∑一世=1nλ一世在一世在一世H. 这称为 Hermitian 矩阵的谱分解。
(7) Hermitian 矩阵的逆矩阵的展开公式一种是
$$\boldsymbol{A}^{-1}=\sum_{i=1}^{n} \frac{\mathrm{I}}{\lambda_{i}} \boldsymbol{u} {i} \ boldsymbol{u} {i}^{H}$$
因此，如果知道 Hermitian 矩阵的特征分解一种，则可以直接得到逆矩阵一种−1使用上面的公式。
(8) 两人份n×n厄米矩阵一种和乙，存在一个酉矩阵，使得磷H一种磷和磷H乙磷都是对角线当且仅当一种乙=乙一种.
(9) 两人份n×n非负定 Hermitian 矩阵一种和乙, 存在一个非奇异矩阵磷以便磷H一种磷和磷H乙磷都是对角线。

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

(1)这⁡(一种−λ乙)=0.
(2) 在∈空值⁡(一种−λ乙)， 和在≠0.

## 机器学习代写|主成分分析作业代写PCA代考|Rayleigh Quotient

r(在)=r(在,C)=在HC在在H在,

(1) 同质性：r(一种在,b在)=br(在,C)∀一种,b≠0.
(2)平移不变性：r(在,C−一种一世)=r(在,C)−一种.
(3) 有界性：自在范围在所有非零向量上，r(在)填充复平面中的一个区域，该区域称为值域C. 这个区域是封闭的、有界的和凸的。如果C=C∗（自伴随矩阵），值域是由极值特征值界定的实区间。
(4) 正交性：在⊥(C−r(在)一世)在.
(5) 最小残差：∀在≠0∧∀标量μ,|(C−r(在)一世)在|≤|(C−μ一世)在|.

Courant-Fischer 定理：让C∈Cn×n是 Hermitian 矩阵，其特征值为λ1≥λ2≥⋯≤λn, 那么它认为对于λķ(1≤ķ≤在) :
λķ=分钟小号,暗淡⁡(小号)=n−ķ+1最大限度在∈小号,在≠0(在HC在在H在)
Courant-Fischer 定理也可以写成
λķ=分钟小号,暗淡⁡(小号)=ķ最大限度在∈小号,在≠0(在HC在在H在)

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