순환 행렬(circulant matrix)과 행렬식

개요

$C_n=\begin{bmatrix}a_0 & a_{1} & \dots & a_{n-2} & a_{n-1} \\a_{n-1} & a_0 & a_{1} & & a_{n-2} \\\vdots & a_{n-1}& a_0 & \ddots & \vdots \\a_{2} & & \ddots & \ddots & a_{1} \\a_{1} & a_{2} & \dots & a_{n-1} & a_0 \\\end{bmatrix}$ 꼴의 행렬

$\left( \begin{array}{c} a_0 \end{array} \right)$

$\left( \begin{array}{cc} a_0 & a_1 \\ a_1 & a_0 \end{array} \right)$

$\left( \begin{array}{ccc} a_0 & a_1 & a_2 \\ a_2 & a_0 & a_1 \\ a_1 & a_2 & a_0 \end{array} \right)$

$\left( \begin{array}{cccc} a_0 & a_1 & a_2 & a_3 \\ a_3 & a_0 & a_1 & a_2 \\ a_2 & a_3 & a_0 & a_1 \\ a_1 & a_2 & a_3 & a_0 \end{array} \right)$

$\left( \begin{array}{ccccc} a_0 & a_1 & a_2 & a_3 & a_4 \\ a_4 & a_0 & a_1 & a_2 & a_3 \\ a_3 & a_4 & a_0 & a_1 & a_2 \\ a_2 & a_3 & a_4 & a_0 & a_1 \\ a_1 & a_2 & a_3 & a_4 & a_0 \end{array} \right)$

행렬식

정리

$C_n$의 행렬식은 다음으로 주어진다 \[\det(C_n)=\prod _{j=0}^{n-1} \sum _{k=0}^{n-1} \omega_{j} ^{ k} a_k\] 여기서 $\omega_j=\exp \left(\frac{2\pi i j}{n}\right)$

예

\[ \begin{array}{l} \begin{aligned} \det(C_1)&=a_0 \\ \det(C_2)&=\left(a_0+a_1\right) \left(a_0+\omega a_1\right) \\ &=a_0^2-a_1^2 \\ \det(C_3)&=\left(a_0+a_1+a_2\right) \left(a_0+\omega ^2 a_1+\omega a_2\right) \left(a_0+\omega a_1+\omega ^2 a_2\right)\\ &=a_0^3+a_1^3+a_2^3 -3 a_1 a_2 a_0\\ \det(C_4)&=\left(a_0+a_1+a_2+a_3\right) \left(a_0+\omega ^3 a_1+\omega ^2 a_2+\omega a_3\right) \left(a_0+\omega ^2 a_1+a_2+\omega ^2 a_3\right) \left(a_0+\omega a_1+\omega ^2 a_2+\omega ^3 a_3\right) \\ &=a_0^4-2 a_2^2 a_0^2-4 a_1 a_3 a_0^2+4 a_2 a_3^2 a_0+4 a_1^2 a_2 a_0-a_1^4+a_2^4-a_3^4+2 a_1^2 a_3^2-4 a_1 a_2^2 a_3 \end{aligned} \end{array} \]

정수 계수 순환 행렬의 예

$$ \begin{array}{l|l} \left( \begin{array}{c} 1 \end{array} \right) & 1 \\ \hline \left( \begin{array}{cc} 1 & 4 \\ 4 & 1 \end{array} \right) & -15 \\ \hline \left( \begin{array}{ccc} 1 & 4 & 9 \\ 9 & 1 & 4 \\ 4 & 9 & 1 \end{array} \right) & 686 \\ \hline \left( \begin{array}{cccc} 1 & 4 & 9 & 16 \\ 16 & 1 & 4 & 9 \\ 9 & 16 & 1 & 4 \\ 4 & 9 & 16 & 1 \end{array} \right) & -62400 \\ \hline \left( \begin{array}{ccccc} 1 & 4 & 9 & 16 & 25 \\ 25 & 1 & 4 & 9 & 16 \\ 16 & 25 & 1 & 4 & 9 \\ 9 & 16 & 25 & 1 & 4 \\ 4 & 9 & 16 & 25 & 1 \end{array} \right) & 9406375 \end{array} $$

역사

메모

수학용어번역

circulant - 대한수학회 수학용어집

매스매티카 파일 및 계산 리소스

https://docs.google.com/file/d/0B8XXo8Tve1cxdWtfcFU1dXRKV3M/edit

사전 형태의 자료

리뷰, 에세이, 강의노트

Irwin Kra and Santiago R. Simanca On Circulant Matrices