2019

Problem 1

A complex square matrix $\mathbf{X}$ is unitary if it holds that $\mathbf{X}{\mathbf{X}}^{\ast }=\mathbf{I}$, where ${\mathbf{X}}^{\ast }$ is the conjugate transpose of $\mathbf{X}$ (also known as the adjoint matrix of $\mathbf{X}$) and $\mathbf{I}$ is the appropriate identity matrix. Let $i$ be the imaginary unit. Answer the following questions.

1. For a positive integer $n$, suppose that $\mathbf{A}$ and $\mathbf{B}$ are unitary matrices of size $n$. Show that the matrix $\mathbf{AB}$ is also unitary.

2. For a positive integer $n$, suppose that $\mathbf{C}$ and $\mathbf{D}$ are real square matrices of size $n$. Let $\mathbf{F}$ be defined as $\mathbf{F}=\mathbf{C}+i\mathbf{D}$ and $\mathbf{G}$ be defined as

   $$\mathbf{G}=\left(\begin{array}{cc}\mathbf{C}& -\mathbf{D}\\ \mathbf{D}& \mathbf{C}\end{array}\right).$$

Show that the matrix $\mathbf{G}$ is orthogonal if and only if the matrix $\mathbf{F}$ is unitary. 3. Find the eigenvalues of the following matrix.

\begin{pmatrix}

1 & 1 & 1 & 1 \

1 & i & -1 & -i \

1 & -1 & 1 & -1 \

1 & -i & -1 & i

\end{pmatrix}

4. For a positive integer $n$, suppose that the $(j, k)$-th element $q_{jk}$ of a square matrix $\mathbf{Q}$ of size $n$ is given by

q_{jk} = \frac{1}{\sqrt{n}} \exp\left(\frac{2\pi i (j-1)(k-1)}{n}\right).

Show that the matrix $\mathbf{Q}$ is unitary. 5. Show that a unitary matrix of size $2$ with determinant $1$ has a form of

\mathbf{H} = \begin{pmatrix}

\exp(i\psi)\cos\theta & \exp(i\psi)\sin\theta \

-\exp(-i\psi)\sin\theta & \exp(-i\psi)\cos\theta

\end{pmatrix},

where $\theta$ and $\psi$ are real numbers. 6. Find the general form of the unitary matrices of size $2$. --- 如果复方阵 $\mathbf{X}$ 满足 $\mathbf{XX^*} = \mathbf{I}$，其中 $\mathbf{X^*}$ 是 $\mathbf{X}$ 的共轭转置（也称为 $\mathbf{X}$ 的伴随矩阵），$\mathbf{I}$ 是适当的单位矩阵，则 $\mathbf{X}$ 是酉矩阵。设 $i$ 为虚数单位。回答以下问题。 1. 对于正整数 $n$，假设 $\mathbf{A}$ 和 $\mathbf{B}$ 是大小为 $n$ 的酉矩阵。证明矩阵 $\mathbf{AB}$ 也是酉矩阵。 2. 对于正整数 $n$，假设 $\mathbf{C}$ 和 $\mathbf{D}$ 是大小为 $n$ 的实方阵。设 $\mathbf{F}$ 定义为 $\mathbf{F} = \mathbf{C} + i\mathbf{D}$，并且 $\mathbf{G}$ 定义为

\mathbf{G} = \begin{pmatrix}

\mathbf{C} & -\mathbf{D} \

\mathbf{D} & \mathbf{C}

\end{pmatrix}.

证明当且仅当矩阵 $\mathbf{F}$ 是酉矩阵时，矩阵 $\mathbf{G}$ 是正交矩阵。 3. 找到以下矩阵的特征值。

\begin{pmatrix}

1 & 1 & 1 & 1 \

1 & i & -1 & -i \

1 & -1 & 1 & -1 \

1 & -i & -1 & i

\end{pmatrix}

4. 对于正整数 $n$，假设方阵 $\mathbf{Q}$ 的第 $(j, k)$ 元素 $q_{jk}$ 由下式给出

q_{jk} = \frac{1}{\sqrt{n}} \exp\left(\frac{2\pi i (j-1)(k-1)}{n}\right).

证明矩阵 $\mathbf{Q}$ 是酉矩阵。 5. 证明一个大小为 $2$ 的酉矩阵且行列式为 $1$ 的矩阵形式为

\mathbf{H} = \begin{pmatrix}

\exp(i\psi)\cos\theta & \exp(i\psi)\sin\theta \

-\exp(-i\psi)\sin\theta & \exp(-i\psi)\cos\theta

\end{pmatrix},

其中 $\theta$ 和 $\psi$ 是实数。 6. 找到大小为 $2$ 的酉矩阵的一般形式。 --- ## Problem 2 The real-valued function $u(x, t)$ is defined for $-\infty < x < \infty$ and $t \geq 0$ with independent variables $x$ and $t$. Consider to solve the partial differential equation

\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}, \tag{2.1}

$$undertheinitialconditions$$

u(x, 0) = \exp(-ax^2) \tag{2.2}

\frac{\partial u}{\partial t}(x, 0) = 0, \tag{2.3}

where $a$ and $c$ are positive real numbers. The imaginary unit is represented by $i$. Answer the following questions. 1. Calculate the following formula by using complex integration

\int_{-\infty}^{\infty} \exp\left(-a(x + id)^2\right) \mathrm{d}x,

where $d$ is a real number. The following equation may be used.

\int_{-\infty}^{\infty} \exp(-x^2) \mathrm{d}x = \sqrt{\pi}

2. The Fourier transform of $u(x, t)$ with respect to $x$, $U(k, t)$, is defined as

U(k, t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} u(x, t) \exp(-ikx) \mathrm{d}x.

You may assume that integration with respect to $x$ and differentiation with respect to $t$ are interchangeable. Also, you may assume that $u(x, t)$ and $\frac{\partial u}{\partial x}(x, t)$ converge to 0 as $x \to \pm \infty$ for an arbitrary $t$. (i) Express the partial differential equation of $U(k, t)$ when $u(x, t)$ satisfies Eq. (2.1). (ii) Show that the solution of (i) takes the following form under the initial condition of Eq. (2.3) using a function $F(k)$ of variable $k$.

U(k, t) = F(k) \cos(kct).

(iii) Furthermore, using the initial condition of Eq. (2.2), determine $U(k, t)$ by finding $F(k)$. The result of question (1) may be used. 3. Find $u(x, t)$ by calculating the inverse Fourier transform of $U(k, t)$ obtained in question (2). The inverse Fourier transform is defined as

u(x, t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} U(k, t) \exp(ikx) \mathrm{d}k.

--- 实值函数 $u(x, t)$ 定义在 $-\infty < x < \infty$ 和 $t \geq 0$，其中 $x$ 和 $t$ 是独立变量。考虑求解偏微分方程

\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}, \tag{2.1}

$$在初始条件下$$

u(x, 0) = \exp(-ax^2) \tag{2.2}

\frac{\partial u}{\partial t}(x, 0) = 0, \tag{2.3}

其中 $a$ 和 $c$ 是正实数。虚数单位由 $i$ 表示。回答以下问题。 1. 使用复积分计算以下公式

\int_{-\infty}^{\infty} \exp\left(-a(x + id)^2\right) \mathrm{d}x,

其中 $d$ 是实数。可以使用以下公式。

\int_{-\infty}^{\infty} \exp(-x^2) \mathrm{d}x = \sqrt{\pi}

2. $u(x, t)$ 关于 $x$ 的傅里叶变换 $U(k, t)$ 定义为

U(k, t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} u(x, t) \exp(-ikx) \mathrm{d}x.

可以假设 $x$ 的积分与 $t$ 的微分可以交换。还可以假设，对于任意 $t$，$u(x, t)$ 和 $\frac{\partial u}{\partial x}(x, t)$ 在 $x \to \pm \infty$ 时收敛于 0。 (i) 在 $u(x, t)$ 满足方程 (2.1) 时，表达 $U(k, t)$ 的偏微分方程。 (ii) 证明在使用函数 $F(k)$ 表示的变量 $k$ 下，(i) 的解在初始条件 (2.3) 下具有以下形式。

U(k, t) = F(k) \cos(kct).

(iii) 此外，使用初始条件 (2.2)，通过找到 $F(k)$ 来确定 $U(k, t)$。可以使用问题 (1) 的结果。 3. 通过计算问题 (2) 中得到的 $U(k, t)$ 的逆傅里叶变换来找到 $u(x, t)$。逆傅里叶变换定义为

u(x, t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} U(k, t) \exp(ikx) \mathrm{d}k.

--- ## Problem 3 Consider a triangle $ABC$ on a plane as shown in the following figure, where the coordinates of the three vertices are $A(1,0)$, $B(0,1)$, and $C(-1,-1)$, respectively. Let $\ell$ be a randomly chosen half line starting from the origin $(0,0)$, that is,

\ell = {(r \cos \Theta, r \sin \Theta) \mid r \geq 0},

where $\Theta$ is a uniformly distributed random variable on the interval $[0, 2\pi)$. Let $Q$ be the point of intersection of the half line $\ell$ and the edges of the triangle $ABC$. Let $(X, Y)$ be the coordinates of $Q$, where $X$ and $Y$ are random variables. Answer the following questions. 1. Find the probability of the event that the point $Q$ is located on the segment $AB$. 2. Show that the expectation of $X$ given the condition that $Q$ is located on $AB$ is $1/2$, where the fact that the triangle $ABC$ is symmetric with respect to the line $y = x$ may be used. 3. Find the probability density function of $X$ given the condition that $Q$ is located on $BC$, by using the change-of-variables formula.

f(x) = g(h(x)) \left| \frac{\mathrm{d}h}{\mathrm{d}x}(x) \right|

where $x$ is an arbitrary real number, $f$ and $g$ are the probability density functions of $X$ and $\Theta$, respectively, and $h$ is the function that satisfies $\Theta = h(X)$. 4. Let $\alpha$ be the expectation of $X$ given the condition that $Q$ is located on $BC$. Calculate $\alpha$ by using the result of question (3). 5. Obtain the expectation $\mu$ of $X$. --- ![[Pasted image 20240709101612.png]] --- 在平面上考虑一个三角形 $ABC$，如图所示，其中三个顶点的坐标分别是 $A(1,0)$、$B(0,1)$ 和 $C(-1,-1)$。设 $\ell$ 为从原点 $(0,0)$ 开始的随机选择的半直线，即

\ell = {(r \cos \Theta, r \sin \Theta) \mid r \geq 0},

其中 $\Theta$ 是在区间 $[0, 2\pi)$ 上均匀分布的随机变量。设 $Q$ 为半直线 $\ell$ 与三角形 $ABC$ 的边的交点。设 $Q$ 的坐标为 $(X, Y)$，其中 $X$ 和 $Y$ 是随机变量。回答以下问题。 1. 求点 $Q$ 位于线段 $AB$ 上的概率。 2. 证明在 $Q$ 位于 $AB$ 上的条件下，$X$ 的期望值为 $1/2$，其中可以使用三角形 $ABC$ 关于直线 $y = x$ 对称这一事实。 3. 使用变量变换公式，求在 $Q$ 位于 $BC$ 上的条件下，$X$ 的概率密度函数。

f(x) = g(h(x)) \left| \frac{\mathrm{d}h}{\mathrm{d}x}(x) \right|

其中 $x$ 是任意实数，$f$ 和 $g$ 分别是 $X$ 和 $\Theta$ 的概率密度函数，$h$ 是满足 $\Theta = h(X)$ 的函数。 4. 设 $\alpha$ 为在 $Q$ 位于 $BC$ 上的条件下，$X$ 的期望值。利用问题 (3) 的结果计算 $\alpha$。 5. 求 $X$ 的期望值 $\mu$。