数学代写|凸优化作业代写Convex Optimization代考|Linear algebra revisited

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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|凸优化作业代写Convex Optimization代考|Vector subspace

A set of vectors $\left{\mathbf{a}{1}, \ldots, \mathbf{a}{k}\right}$ is said to be linearly independent if the following equality holds only when $\alpha_{1}=\alpha_{2}=\cdots=\alpha_{k}=0$,
$$\alpha_{1} \mathbf{a}{1}+\alpha{2} \mathbf{a}{2}+\cdots+\alpha{k} \mathbf{a}{k}=\mathbf{0} .$$ A set of vectors $\left{\mathbf{a}{1}, \ldots, \mathbf{a}{k}\right}$ is said to be linearly dependent if any one of the vectors from the set is a linear combination of the remaining vectors or if one of the vectors is a zero vector. The vector set $\left{\mathbf{a}{1}, \ldots, \mathbf{a}_{k}\right}$ is linearly dependent if it is not linearly independent and vice versa.

A subset $V$ of $\mathbb{R}^{n}$ is called a subspace of $\mathbb{R}^{n}$ if $V$ is closed under the operations of vector addition and scalar multiplication (i.e., $\alpha \mathbf{v}{1}+\beta \mathbf{v}{2} \in V$ for all $\alpha, \beta \in \mathbb{R}$ and $\mathbf{v}{1}, \mathbf{v}{2} \in V$ ). Note that every subspace must contain the zero vector.

Let $\mathbf{a}{1}, \mathbf{a}{2}, \ldots, \mathbf{a}{k}$ be arbitrary vectors in $\mathbb{R}^{n}$. The set of all their linear combinations is called the span of $\mathbf{a}{1}, \mathbf{a}{2}, \ldots, \mathbf{a}{k}$ and is denoted as
$$\operatorname{span}\left[\mathbf{a}{1}, \ldots, \mathbf{a}{k}\right]=\left{\sum_{i=1}^{k} \alpha_{i} \mathbf{a}{i} \mid \alpha{1}, \alpha_{2}, \ldots, \alpha_{k} \in \mathbb{R}\right}$$
Note that the span of any set of vectors is a subspace.
Given a subspace $V$, any set of linearly independent vectors $\left{\mathbf{a}{1}, \ldots, \mathbf{a}{k}\right} \subset V$ such that $V=\operatorname{span}\left[\mathbf{a}{1}, \ldots, \mathbf{a}{k}\right]$ is referred to as a basis of the subspace $V$. All bases of a subspace $V$ contain the same number of vectors and this number is called the dimension of $V$ and is denoted as $\operatorname{dim}(V)$. Any vector in $V$ can be represented uniquely by a linear combination of the vectors of any basis of $V$.
Range space, null space, and orthogonal projection
Let $\mathbf{A}=\left[\mathbf{a}{1}, \ldots, \mathbf{a}{n}\right] \in \mathbb{R}^{m \times n}$. The range space or image (also a subspace) of the matrix $\mathbf{A}$ is defined as
$$\mathcal{R}(\mathbf{A})=\left{\mathbf{y} \in \mathbb{R}^{m} \mid \mathbf{y}=\mathbf{A} \mathbf{x}, \mathbf{x} \in \mathbb{R}^{n}\right}=\operatorname{span}\left[\mathbf{a}{1}, \ldots, \mathbf{a}{n}\right],$$
and the rank of $\mathbf{A}$, denoted as $\operatorname{rank}(\mathbf{A})$, is the maximum number of independent columns (or independent rows) of $\mathbf{A}$. In fact, $\operatorname{dim}(\mathcal{R}(\mathbf{A}))=\operatorname{rank}(\mathbf{A})$. Some facts about matrix rank are as follows:

• If $\mathbf{A} \in \mathbb{R}^{m \times k}, \mathbf{B} \in \mathbb{R}^{k \times n}$, then
$$\operatorname{rank}(\mathbf{A})+\operatorname{rank}(\mathbf{B})-k \leq \operatorname{rank}(\mathbf{A B}) \leq \min {\operatorname{rank}(\mathbf{A}), \operatorname{rank}(\mathbf{B})} .$$
• If $\mathbf{A} \in \mathbb{R}^{m \times m}, \mathbf{C} \in \mathbb{R}^{n \times n}$ are both nonsingular, and $\mathbf{B} \in \mathbb{R}^{m \times n}$, then
$$\operatorname{rank}(\mathbf{B})=\operatorname{rank}(\mathbf{A B})=\operatorname{rank}(\mathbf{B C})=\operatorname{rank}(\mathbf{A B C}) .$$

数学代写|凸优化作业代写Convex Optimization代考|Matrix determinant and inverse

Let $\mathbf{A}=\left{a_{i, j}\right}_{n \times n} \in \mathbb{R}^{n \times n}$ and $\mathcal{A}{i j} \in \mathbb{R}^{(n-1) \times(n-1)}$ be the submatrix of $\mathbf{A}$ by deleting the $i$ th row and $j$ th column of $\mathbf{A}$. Then the detrminant of $\mathbf{A}$ is defined as $$\operatorname{det}(\mathbf{A})= \begin{cases}\sum{j=1}^{n} a_{i j} \cdot(-1)^{i+j} \operatorname{det}\left(\mathcal{A}{i j}\right), & \forall i \in{1, \ldots, n} \ \sum{i=1}^{n} a_{i j} \cdot(-1)^{i+j} \operatorname{det}\left(\mathcal{A}{i j}\right), & \forall j \in{1, \ldots, n}\end{cases}$$ which is called the cofactor expansion since the term in each summation $(-1)^{i+j} \operatorname{det}\left(\mathcal{A}{i j}\right)$ is the $(i, j)$ th cofactor of $\mathbf{A}$.
The inverse of $\mathbf{A}$ is defined as
$$\mathbf{A}^{-1}=\frac{1}{\operatorname{det}(\mathbf{A})} \cdot \operatorname{adj}(\mathbf{A})$$
where $\operatorname{adj}(\mathbf{A}) \in \mathbb{R}^{n \times n}$ denotes the adjoint matrix of $\mathbf{A}$ with the $(j, i)$ th element given by
$${\operatorname{adj}(\mathbf{A})}_{j i}=(-1)^{i+j} \operatorname{det}\left(\boldsymbol{A}{i j}\right) .$$ A useful matrix inverse identity, called the Woodbury identity, is given by Some other useful matrix inverse identities and matrix determinants are given as follows: \begin{aligned} (\mathbf{A B})^{-1} &=\mathbf{B}^{-1} \mathbf{A}^{-1} \ \left(\mathbf{A}^{T}\right)^{-1} &=\left(\mathbf{A}^{-1}\right)^{T} \ \operatorname{det}\left(\mathbf{A}^{T}\right) &=\operatorname{det}(\mathbf{A}) \ \operatorname{det}\left(\mathbf{A}^{-1}\right) &=1 / \operatorname{det}(\mathbf{A}) \ \operatorname{det}(\mathbf{A B}) &=\operatorname{det}(\mathbf{A}) \cdot \operatorname{det}(\mathbf{B}) \ \operatorname{det}\left(\mathbf{I}{n}+\mathbf{u} \mathbf{v}^{T}\right) &=1+\mathbf{u}^{T} \mathbf{v}, \mathbf{u}, \mathbf{v} \in \mathbb{R}^{n} . \end{aligned}
Note that $\mathbf{u v}^{T}$ in (1.85) is a rank-1 asymmetric $n \times n$ matrix with one nonzero eigenvalue equal to $\mathbf{u}^{T} \mathbf{v}$ and a corresponding eigenvector $\mathbf{u}$ (to be introduced in Subsection 1.2.5).

数学代写|凸优化作业代写Convex Optimization代考|Positive definiteness and semidefiniteness

An $n \times n$ real symmetric matrix $\mathbf{M}$ is positive definite (PD) (i.e., $\mathbf{M} \in \mathbb{S}{++}^{n}$ ) if $\mathbf{z}^{T} \mathbf{M z}>0$ for any nonzero vector $\mathbf{z} \in \mathbb{R}^{n}$, where $\mathbf{z}^{T}$ denotes the transpose of $\mathbf{z}$. $\mathbf{M} \succ \mathbf{0}$ is also used to denote that $\mathbf{M}$ is a $\mathrm{PD}$ matrix. For complex matrices, this definition becomes: a Hermitian matrix $\mathbf{M}=\mathbf{M}^{H}=\left(\mathbf{M}^{*}\right)^{T} \in \mathbb{H}{++}^{n}$ is positive definite if $\mathbf{z}^{H} \mathbf{M z}>0$ for any nonzero complex vector $\mathbf{z} \in \mathbb{C}^{n}$, where $\mathbf{z}^{H}$ denotes the conjugate transpose of $z$.

Remark $1.16$ An $n \times n$ real symmetric matrix $\mathbf{M}$ is said to be positive semidefinite $(\mathrm{PSD})$ (i.e., $\left.\mathbf{M} \in \mathrm{S}{+}^{n}\right)$ and negative definite if $\mathbf{z}^{T} \mathbf{M z} \geq 0$ and $\mathbf{z}^{T} \mathbf{M z}<0$, respectively, for any nonzero vector $z \in \mathbb{R}^{n}$. An $n \times n$ Hermitian PSD matrix can be defined similarly, and $\mathbf{M} \succeq \mathbf{0}$ is also used to denote that $\mathbf{M}$ is a PSD matrix. A real symmetric $n \times n$ matrix $\mathbf{X}$ is called indefinite if there exist $\mathbf{z}{1}, \mathbf{z}{2} \in \mathbb{R}^{n}$ such that $\mathbf{z}{1}^{T} \mathbf{X} \mathbf{z}{1}>0$ and $\mathbf{z}{2}^{T} \mathbf{X} \mathbf{z}{2}<0$; so is the case of indefinite $n \times n$ Hermitian matrix $\mathbf{X}$ for which $\mathbf{z}{1}^{H} \mathbf{X} \mathbf{z}{1}>0$ and $\mathbf{z}{2}^{H} \mathbf{X} \mathbf{z}{2}<0$ where $\mathbf{z}{1}, \mathbf{z}_{2} \in \mathbb{C}^{n}$.

Remark 1.17 The mathematical definitions of PD and PSD matrices do not require the matrix to be symmetric or Hermitian. However, we only concentrate on the real symmetric matrices or complex Hermitian matrices because this is the case in most practical applications by our experience on one hand, and a lot of available mathematical results on symmetric or Hermitian matrices can be utilized in the development and analysis of convex optimization algorithms on the other hand.

数学代写|凸优化作业代写Convex Optimization代考|Vector subspace

\operatorname{span}\left[\mathbf{a}{1}, \ldots, \mathbf{a}{k}\right]=\left{\sum_{i=1}^{k} \alpha_{i } \mathbf{a}{i} \mid \alpha{1}, \alpha_{2}, \ldots, \alpha_{k} \in \mathbb{R}\right}\operatorname{span}\left[\mathbf{a}{1}, \ldots, \mathbf{a}{k}\right]=\left{\sum_{i=1}^{k} \alpha_{i } \mathbf{a}{i} \mid \alpha{1}, \alpha_{2}, \ldots, \alpha_{k} \in \mathbb{R}\right}

\mathcal{R}(\mathbf{A})=\left{\mathbf{y} \in \mathbb{R}^{m} \mid \mathbf{y}=\mathbf{A} \mathbf{x} , \mathbf{x} \in \mathbb{R}^{n}\right}=\operatorname{span}\left[\mathbf{a}{1}, \ldots, \mathbf{a}{n}\对]，\mathcal{R}(\mathbf{A})=\left{\mathbf{y} \in \mathbb{R}^{m} \mid \mathbf{y}=\mathbf{A} \mathbf{x} , \mathbf{x} \in \mathbb{R}^{n}\right}=\operatorname{span}\left[\mathbf{a}{1}, \ldots, \mathbf{a}{n}\对]，

• 如果一种∈R米×ķ,乙∈Rķ×n， 然后
秩⁡(一种)+秩⁡(乙)−ķ≤秩⁡(一种乙)≤分钟秩⁡(一种),秩⁡(乙).
• 如果一种∈R米×米,C∈Rn×n都是非奇异的，并且乙∈R米×n， 然后
秩⁡(乙)=秩⁡(一种乙)=秩⁡(乙C)=秩⁡(一种乙C).

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