# 循环矩阵

## 定义

${\displaystyle C={\begin{bmatrix}c_{0}&c_{n-1}&c_{n-2}&\dots &c_{1}\\c_{1}&c_{0}&c_{n-1}&c_{2}\\c_{2}&c_{1}&c_{0}&c_{n-1}\\\vdots &&&\ddots &\vdots \\c_{n-1}&c_{n-2}&c_{n-3}&\dots &c_{0}\end{bmatrix}}}$

${\displaystyle n\times n}$ 矩阵 C 就是循环矩阵

## 特性

${\displaystyle \lambda _{j}=c_{0}+c_{n-1}\omega _{j}+c_{n-2}\omega _{j}^{2}+\ldots +c_{1}\omega _{j}^{n-1},\qquad j=0,1,\ldots ,n-1.}$

## 对称循环矩阵

${\displaystyle C={\begin{bmatrix}c_{0}&c_{1}&\dots &c_{2}&c_{1}\\c_{1}&c_{0}&c_{1}&&c_{2}\\\vdots &c_{1}&c_{0}&\ddots &\vdots \\c_{2}&&\ddots &\ddots &c_{1}\\c_{1}&c_{2}&\dots &c_{1}&c_{0}\\\end{bmatrix}}.}$

${\displaystyle \lambda _{j}=c_{0}+2c_{1}\Re \omega _{j}+2c_{2}\Re \omega _{j}^{2}+\ldots +2c_{n/2-1}\Re \omega _{j}^{n/2-1}+c_{n/2}\omega _{j}^{n/2}}$

${\displaystyle n}$为奇数时为

${\displaystyle \lambda _{j}=c_{0}+2c_{1}\Re \omega _{j}+2c_{2}\Re \omega _{j}^{2}+\ldots +2c_{(n-1)/2}\Re \omega _{j}^{(n-1)/2}}$

## 用循环矩阵来解线性方程

${\displaystyle \mathbf {C} \mathbf {x} =\mathbf {b} }$

${\displaystyle \mathbf {c} *\mathbf {x} =\mathbf {b} }$

${\displaystyle {\mathcal {F}}_{n}(\mathbf {c} *\mathbf {x} )={\mathcal {F}}_{n}(\mathbf {c} ){\mathcal {F}}_{n}(\mathbf {x} )={\mathcal {F}}_{n}(\mathbf {b} )}$

${\displaystyle \mathbf {x} ={\mathcal {F}}_{n}^{-1}\left[\left({\frac {({\mathcal {F}}_{n}(\mathbf {b} ))_{\nu }}{({\mathcal {F}}_{n}(\mathbf {c} ))_{\nu }}}\right)_{\nu \in \mathbf {Z} }\right].}$