## 每週問題 February 13, 2017

Let $A$ be a unitary matrix, i.e., $A^\ast=A^{-1}$. Prove that

$\displaystyle \lim_{m\to\infty}\frac{1}{m}\sum_{k=0}^{m-1}A^k\mathbf{x}=P\mathbf{x}$,

where $P$ is the Hermitian projection matrix onto $N(I-A^\ast)$.

## 每週問題 February 6, 2017

Let $A$ be an $n\times n$ real symmetric positive definite matrix. Prove that

$\displaystyle \int_{-\infty}^\infty\cdots\int_{-\infty}^\infty e^{-\mathbf{x}^TA\mathbf{x}}dx_1\cdots dx_n=\pi^{n/2}(\det A)^{-1/2}$,

where $\mathbf{x}=(x_1,\ldots,x_n)^T$.

## 每週問題 January 30, 2017

Prove that the rank of a real skew-symmetric matrix is an even number.

## 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

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

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