1. 复数矩阵

复向量

$$Z=\left[\begin{array}{c}{z}_1\\ z_2\\ z_3\\ z_4\\ \cdots \end{array}\right]$$
复向量的模长
$$|z|={\overline{z}}^{\top }z=\left[\begin{array}{c}{\overline{z}}_1{\overline{z}}_2{\overline{z}}_3\end{array}\right]\left[\begin{array}{c}{z}_1\\ z_2\\ z_3\end{array}\right]$$

内积
$$y^{\top }x=\left[\begin{array}{c}{\overline{y}}_1{\overline{y}}_2{\overline{y}}_3\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]$$
实对称矩阵
$$A=A^{\top }$$
复对称矩阵
$$A={\overline{A}}^{\top }$$

如
$$\left[\begin{array}{cc}2& 3-i\\ 3+i& 5\end{array}\right]$$
垂直
$$\begin{gathered} q_1{q}_2{q}_3\cdots q_n \\ {\overline{q}}_i^{\top }{q}_j=\{\begin{array}{l}0i\ne j\\ 1i=j\end{array} \end{gathered}$$

复正交矩阵
$${\overline{Q}}^{\top }Q=I$$

2. 快速傅里叶变换

$$F_n=\left[\begin{array}{ccccc}1& 1& 1& \cdots & 1\\ 1& \omega & {\omega }^2& \cdots & {\omega }^{n-1}\\ ⋮& \cdots & & & \\ 1& {\omega }^{n-1}& {\omega }^{2(n-1)}& \cdots & {\omega }^{(n-1)(n-1)}\end{array}\right]$$

$$\begin{gathered} F_n(i,j)=w^{ij},{\omega }^n=1 \\ \omega =e^{i\frac{2\pi }{n}}=\cos \frac{2\pi }{n}+i\sin \frac{2\pi }{n} \end{gathered}$$

$$F_4=\left[\begin{array}{cccc}1& 1& 1& 1\\ 1& i& i^2& i^3\\ 1& i^2& i^4& i^6\\ 1& i^3& i^6& i^9\end{array}\right]=\frac{\sqrt[]{2}}{2}\left[\begin{array}{cccc}1& 1& 1& 1\\ 1& i& -1& -i\\ 1& -1& 1& -1\\ 1& -i& 1& i\end{array}\right]$$
矩阵各列正交。

$${\omega }^n=e^{\frac{i\times 2\pi }{n}}$$

$$w^n\ast w^n=w^{2n}$$

对于$n=64$,可以化为

$$[F_{64}]=\left[\begin{array}{cc}I& D\\ I& -D\end{array}\right]\left[\begin{array}{cc}{F}_{32}& 0\\ 0& F_{32}\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ \cdots & & \end{array}\right]$$

$D$是一个对角矩阵
$$D=\left[\begin{array}{ccc}1& & \\ & w& \\ & & w^{n-1}\end{array}\right]$$

对于$F_{32}$可以继续做这样的分解，直到分解成$F_1$

即
$$F_n=DMP$$
$M$为分解矩阵，分解成两个小规模的矩阵。
$$M=\left[\begin{array}{cc}{F}_{n/2}& 0\\ 0& F_{n/2}\end{array}\right]$$

矩阵$P$为奇偶位次置换矩阵。

$todo$