## 高斯混合模型與最大期望算法

$\displaystyle \mathcal{L}(\boldsymbol{\theta}|\mathcal{X})=p(\mathcal{X}|\boldsymbol{\theta})=\prod_{i=1}^np(\mathbf{x}_i|\boldsymbol{\theta})$

$\displaystyle \boldsymbol{\theta}^\ast=\arg\max_{\boldsymbol{\theta}}\mathcal{L}(\boldsymbol{\theta}|\mathcal{X})$

## 每週問題 January 16, 2017

Let $A$ and $B$ be $n\times n$ real symmetric matrices, and $C(\lambda)=\lambda A+(1-\lambda)B$, $\lambda\in\mathbb{R}$. If there exists a $\lambda\in[0,1]$ such that $C(\lambda)$ is a positive semidefinite matrix and $\hbox{null}\,C(\lambda)=\hbox{null}\,A\cap \hbox{null}\,B$, then there exists a nonsingular matrix $P$ such that both $P^TAP$ and $P^TBP$ are diagonal. Note that $\hbox{null}\,X$ denotes the nullspace of $X$.

## 因素分析

$\displaystyle \mathbf{z}=W^T(\mathbf{x}-\boldsymbol{\mu})$

$\displaystyle \mathbf{x}=\boldsymbol{\mu}+F\mathbf{z}+\boldsymbol{\epsilon}$

• 因素分析如何描述多隨機變數的產生？
• 如何估計因素分析的模型參數？
• 因素分析如何解釋隱藏因素的涵義？
• 因素分析如何應用於降維？
• 因素分析與主成分分析有哪些相同與相異的性質？

## 主成分分析與低秩矩陣近似

$X=\begin{bmatrix} \mathbf{x}_1^T\\ \vdots\\ \mathbf{x}_n^T \end{bmatrix}=\begin{bmatrix} x_{11}&\cdots&x_{1p}\\ \vdots&\ddots&\vdots\\ x_{n1}&\cdots&x_{np} \end{bmatrix}$

## 每週問題 January 9, 2017

Let $\mathcal{V}$ and $\mathcal{W}$ be two vector spaces over the same field. Suppose $F$ and $G$ are two linear transformations $\mathcal{V}\to \mathcal{W}$ such that for every $\mathbf{x}\in\mathcal{V}$, $G(\mathbf{x})$ is s scalar multiple (depending on $\mathbf{x}$) of $F(\mathbf{x})$. Prove that $G$ is a scalar multiple of $F$.

## 每週問題 January 2, 2017

$A$ 是一個二階方陣且 $\hbox{trace}A=0$，證明存在一個么正 (unitary) 矩陣 $U$ 使得 $U^\ast AU$ 的主對角元為零。

Let $A$ be a $2\times 2$ matrix and $\hbox{trace}A=0$. Show that there exists a unitary matrix $U$ such that the diagonal elements of $U^\ast AU$ are equal to zero.

## 每週問題 December 26, 2016

Let $A$ be an $n\times n$ real matrix. Show that $A$ and $A^{-1}$ have all elements nonnegative if and only if each row and each column of $A$ has exactly one positive element and the rest of the elements are zeros.

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## 約束最小平方問題

$A$ 為一個 $m\times n$ 階實矩陣，$\mathbf{b}\in\mathbb{R}^m$。如果線性方程 $A\mathbf{x}=\mathbf{b}$ 是不一致的 (即不存在解)，實務的作法是將線性方程問題改為最小平方近似問題：

$\displaystyle \min_{\mathbf{x}}\Vert A\mathbf{x}-\mathbf{b}\Vert^2$

$\displaystyle \hat{\mathbf{x}}=(A^TA)^{-1}A^T\mathbf{b}$

$\displaystyle \min_{C\mathbf{x}=\mathbf{d}}\Vert A\mathbf{x}-\mathbf{b}\Vert^2$

$\displaystyle \min_{\Vert\mathbf{x}\Vert=d}\Vert A\mathbf{x}-\mathbf{b}\Vert^2$

## 每週問題 December 19, 2016

Let $\mathcal{V}$ be a vector space, $\dim\mathcal{V}=n$, and let $\mathbf{x}_1,\ldots,\mathbf{x}_m\in\mathcal{V}$. Prove that if $m\ge n+2$, then there exist scalars $c_1,\ldots,c_m$ not all of them equal to zero such that $\sum_{i=1}^mc_i\mathbf{x}_i=\mathbf{0}$ and $\sum_{i=1}^mc_i=0$.

## 每週問題 December 12 , 2016

An $n\times n$ matrix is said to be a semi-magic matrix if the sums of the rows and columns are all equal. Show that a semi-magic matrix $A$ can be decomposed as $A=B+C$ such that for integer $k\ge 1$,

$A^k=B^k+C^k$.

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