## 線性基函數模型

$\displaystyle y(\mathbf{x};\mathbf{w})=w_0+w_1x_1+\cdots+w_dx_d$

$\displaystyle y(\mathbf{x};\mathbf{w})=w_0+w_1\phi_1(\mathbf{x})+\cdots+w_{m-1}\phi_{m-1}(\mathbf{x})$

$\displaystyle y(\mathbf{x};\mathbf{w})=\sum_{j=0}^{m-1}w_j\phi_j(\mathbf{x})=\mathbf{w}^T\boldsymbol{\phi}(\mathbf{x})$

## 每週問題 January 23, 2017

Prove that if $A$ is a real symmetric positive definite then $\hbox{adj}A$ is also a symmetric positive definite matrix.

## 2017 年大學學測的線性代數問題

$a_1,\ldots,a_9$ 為等差數列，且 $k$ 為實數，若方程組

\left\{\begin{aligned} a_1x-a_2y+2a_3z&=k+1\\ a_4x-a_5y+2a_6z&=-k-5\\ a_7x-a_8y+2a_9z&=k+9 \end{aligned}\right.

〈迷神〉開頭寫道：

## 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.