命名约定

$a, b, c \in \mathbb{C}, \mathbf{a}=(\mathbf{Re}(a), \mathbf{Im}(a)), a^*=\bar{a}, \alpha=\arg a$.

分解 I

$\mathbf{Re}(z) = \dfrac{z+z^*}{2}$

$\mathbf{Im}(z) = \dfrac{z-z^*}{2\mathrm{i}}$

点积

$$\begin{aligned} (z_1+z_2)(z_1^*+z_2^*) &= (\mathbf{z_1}+\mathbf{z_2})^2 \\ z_1z_1^*+z_2z_2^*+z_1z_2^*+z_1^*z_2 &= \mathbf{z_1}^2+\mathbf{z_2}^2+2\mathbf{z_1}\cdot\mathbf{z_2} \\ \mathbf{z_1}\cdot\mathbf{z_2} &= \frac{z_1z_2^*+z_1^*z_2}{2} \end{aligned}$$

当然也有 $\mathbf{z_1}\cdot\mathbf{z_2} = \mathbf{Re}(z_1^*z_2) = \mathbf{Re}(z_2^*z_1)$.

另一种优雅的证明方法见下。

叉积

将二维叉积定义为 $\mathbf{a}\times\mathbf{b}=\det(\mathbf{a}, \mathbf{b})$.

$$\mathbf{z_1}\times\mathbf{z_2} = \frac{z_1z_2^* - z_1^*z_2}{2} = \mathbf{Im}(z_1^*z_2) = -\mathbf{Im}(z_2^*z_1)$$

证明 $a^*b=|a|e^{-i\alpha}|b|e^{i\beta}=|a||b|e^{i(\beta-\alpha)}=|a||b|\cos(\beta-\alpha)+i|a||b|\sin(\beta-\alpha)=\mathbf{a}\cdot\mathbf{b}+\mathrm{i}\mathbf{a}\times\mathbf{b}$

线性变换

可以由点积表示。

$$\left[ \begin{matrix}\mathbf{a}\\\mathbf{b}\end{matrix} \right] ~ \mathbf{c} = \left[ \begin{matrix}\mathbf{a}\cdot\mathbf{c} \\ \mathbf{b}\cdot\mathbf{c}\end{matrix} \right]$$

分解 II

$\mathbf{i}$ 易混淆，我们用 $\mathbf{u}, \mathbf{v}$ 表示基底。

设 $u=a+b\mathrm{i}, v=c+d\mathrm{i}, w=m+n\mathrm{i}$.

若 $w=xu+yv$，则有

$$\begin{cases} ax+cy=m \\ bx+dy=n \end{cases}$$

由克莱默法则

$$\begin{cases} x = \frac{\begin{vmatrix} m&c\\n&d \end{vmatrix}} {\begin{vmatrix} a&c\\b&d \end{vmatrix}} = \dfrac{\mathbf{w}\times\mathbf{v}}{\mathbf{u}\times\mathbf{v}} \\ y = \frac{\begin{vmatrix} a&m\\b&n \end{vmatrix}} {\begin{vmatrix} a&c\\b&d \end{vmatrix}} = \dfrac{\mathbf{u}\times\mathbf{w}}{\mathbf{u}\times\mathbf{v}} \end{cases}$$

$$x+yi = \frac{\mathbf{Im}(w^*v) + i\mathbf{Im}(u^*w)}{\mathbf{Im}(u^*v)}$$

$$x+yi = \frac{w^*v-wv^* + i(u^*w-uw^*)}{u^*v-uv^*}$$

Ex：走近新定义

$$a*b \mathop{=}\limits^\triangle a^*b$$

$$\begin{aligned} a*b=(b*a)^* &~~~~\text{（共轭交换律）} \\ (a*b)*c=a^* *(b*c) &~~~~\text{（共轭结合律，即左乘左外共轭律）}\\ a*(b*c) = b*(a*c) &~~~~\text{（左乘右外互通律）}\\ a*(b*c) = (a^* *b) * c &~~~~\text{（左乘左外共轭律）}\\ (a*b)*c = b*(a^* *c) = b*(ac) &~~~~\text{（左乘左右共轭律）}\\ a(b*c) = b*(ac)=(a*b)*c &~~~~\text{（左乘右外互通律/跨运算结合律）}\\ a(b*c) = (a^* b) * c &~~~~\text{（左乘左外共轭律）}\\ (ab)*c = b*(a^*c) &~~~~\text{（左乘左右共轭律/跨运算结合律）}\\ (a \pm b)*c = a*c \pm b*c &~~~~\text{（右分配律）}\\ a*(b \pm c) = a*b \pm a*c &~~~~\text{（左分配律）}\\ a^{-1} = \frac{1}{a^*} &~~~~\text{（单位圆反演逆元）}\\ aa^{-1}=a^{-1}a=1 &~~~~\text{（左右逆元相等）}\\ \begin{aligned} a*a=|a|^2 \\ a*(a\mathrm{i})=|a|^2\mathrm{i} \end{aligned} &~~~~\text{（模长平方）} \end{aligned}$$