"순환 행렬(circulant matrix)과 행렬식"의 두 판 사이의 차이
Pythagoras0 (토론 | 기여) |
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| (사용자 2명의 중간 판 22개는 보이지 않습니다) | |||
| 1번째 줄: | 1번째 줄: | ||
| − | + | ==개요== | |
| − | + | <math>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}</math> 꼴의 행렬 | |
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<math>\left( \begin{array}{c} a_0 \end{array} \right)</math> | <math>\left( \begin{array}{c} a_0 \end{array} \right)</math> | ||
| − | <math>\left( \begin{array}{cc} a_1 | + | <math>\left( \begin{array}{cc} a_0 & a_1 \\ a_1 & a_0 \end{array} \right)</math> |
| − | <math>\left( \begin{array}{ccc} a_1 & a_2 | + | <math>\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)</math> |
| − | <math>\left( \begin{array}{cccc} a_1 & a_2 & a_3 | + | <math>\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)</math> |
| − | <math>\left( \begin{array}{ccccc} a_1 & a_2 & a_3 & a_4 | + | <math>\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)</math> |
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| + | ==대각화== | ||
| + | * [[이산 푸리에 변환]]의 행렬을 이용하여 대각화 할 수 있다 | ||
| + | ;정리 | ||
| + | primitive인 n-단위근 <math>\omega</math>에 대하여, <math>\omega_j:=\omega^j</math>라 두자. 벡터 <math>(\omega_j^{k})_{0\leq k \leq n-1}</math>는 순환 행렬 <math>C_n</math>의 고유벡터이며, 고유값은 <math>\sum_{k=0}^{n-1}\omega_{j}^{k}a_k</math>로 주어진다 | ||
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| − | < | + | ===예=== |
| + | * <math>\omega=e^{2\pi i/4}=i</math> | ||
| + | * 행렬 <math>\mathcal{F}_4</math>를 다음과 같이 정의하자 | ||
| + | :<math>\mathcal{F}_4=\left( | ||
| + | \begin{array}{cccc} | ||
| + | 1 & 1 & 1 & 1 \\ | ||
| + | 1 & \omega & \omega ^2 & \omega ^3 \\ | ||
| + | 1 & \omega ^2 & 1 & \omega ^2 \\ | ||
| + | 1 & \omega ^3 & \omega ^2 & \omega \\ | ||
| + | \end{array} | ||
| + | \right) | ||
| + | </math> | ||
| + | * 다음이 성립한다 | ||
| + | :<math> | ||
| + | \mathcal{F}_4^{-1}C_4\mathcal{F}_4=\left( | ||
| + | \begin{array}{cccc} | ||
| + | a_0+a_1+a_2+a_3 & 0 & 0 & 0 \\ | ||
| + | 0 & a_3 \omega ^3+a_2 \omega ^2+a_1 \omega +a_0 & 0 & 0 \\ | ||
| + | 0 & 0 & a_1 \omega ^2+a_3 \omega ^2+a_0+a_2 & 0 \\ | ||
| + | 0 & 0 & 0 & a_1 \omega ^3+a_2 \omega ^2+a_3 \omega +a_0 \\ | ||
| + | \end{array} | ||
| + | \right) | ||
| + | </math> | ||
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| − | + | ==행렬식== | |
| + | ;정리 | ||
| + | <math>C_n</math>의 행렬식은 다음으로 주어진다 | ||
| + | :<math>\det(C_n)=\prod _{j=0}^{n-1} \sum _{k=0}^{n-1} \omega_{j} ^{ k} a_k</math> 여기서 <math>\omega_j=\exp \left(\frac{2\pi i j}{n}\right)</math> | ||
| − | + | ===예=== | |
| + | :<math> | ||
| + | \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} | ||
| + | </math> | ||
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| − | + | ===정수 계수 순환 행렬의 예=== | |
| + | :<math> | ||
| + | \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} | ||
| + | </math> | ||
| − | + | ==역사== | |
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* http://www.google.com/search?hl=en&tbs=tl:1&q= | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
| − | * [[ | + | * [[수학사 연표]] |
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| + | ==메모== | ||
* http://mathoverflow.net/questions/104368/rational-solutions-to-x3-y3-z3-3xyz-1 | * http://mathoverflow.net/questions/104368/rational-solutions-to-x3-y3-z3-3xyz-1 | ||
| − | * | + | * http://videolectures.net/mit18085f07_strang_lec23/ |
| − | + | * Yilmaz, Nazmiye, Yasin Yazlik, and Necati Taskara. “On the Properties of Circulant Matrices Involving Generalized Tribonacci and Generalized Tribonacci-Lucas Numbers.” arXiv:1407.3925 [math], July 15, 2014. http://arxiv.org/abs/1407.3925. | |
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| − | + | ==관련된 항목들== | |
* [[유한아벨군과 이산푸리에변환]] | * [[유한아벨군과 이산푸리에변환]] | ||
| + | * [[라그랑지 resolvent]] | ||
| + | * [[원분체 (cyclotomic field)]] | ||
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| − | + | ==수학용어번역== | |
| + | * {{수학용어집|url=circulant}} | ||
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| − | + | ==매스매티카 파일 및 계산 리소스== | |
| + | * https://docs.google.com/file/d/0B8XXo8Tve1cxdWtfcFU1dXRKV3M/edit | ||
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| − | + | ==사전 형태의 자료== | |
* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
* http://en.wikipedia.org/wiki/circulant_matrix | * http://en.wikipedia.org/wiki/circulant_matrix | ||
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| − | + | ==리뷰, 에세이, 강의노트== | |
| − | + | * Irwin Kra and Santiago R. Simanca [http://www.ams.org/notices/201203/rtx120300368p.pdf On Circulant Matrices] | |
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| − | + | ==관련논문== | |
| + | * Kushel, Olga, and Mikhail Tyaglov. “Circulants and Critical Points of Polynomials.” arXiv:1512.07983 [math], December 25, 2015. http://arxiv.org/abs/1512.07983. | ||
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| − | + | [[분류:선형대수학]] | |
| + | [[분류:행렬식]] | ||
| − | * | + | ==메타데이터== |
| − | + | ===위키데이터=== | |
| − | * | + | * ID : [https://www.wikidata.org/wiki/Q245457 Q245457] |
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LOWER': 'circulant'}, {'LEMMA': 'matrix'}] | ||
2021년 2월 17일 (수) 04:49 기준 최신판
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
\(\left( \begin{array}{c} a_0 \end{array} \right)\)
\(\left( \begin{array}{cc} a_0 & a_1 \\ a_1 & a_0 \end{array} \right)\)
\(\left( \begin{array}{ccc} a_0 & a_1 & a_2 \\ a_2 & a_0 & a_1 \\ a_1 & a_2 & a_0 \end{array} \right)\)
\(\left( \begin{array}{cccc} a_0 & a_1 & a_2 & a_3 \\ a_3 & a_0 & a_1 & a_2 \\ a_2 & a_3 & a_0 & a_1 \\ a_1 & a_2 & a_3 & a_0 \end{array} \right)\)
\(\left( \begin{array}{ccccc} a_0 & a_1 & a_2 & a_3 & a_4 \\ a_4 & a_0 & a_1 & a_2 & a_3 \\ a_3 & a_4 & a_0 & a_1 & a_2 \\ a_2 & a_3 & a_4 & a_0 & a_1 \\ a_1 & a_2 & a_3 & a_4 & a_0 \end{array} \right)\)
대각화
- 이산 푸리에 변환의 행렬을 이용하여 대각화 할 수 있다
- 정리
primitive인 n-단위근 \(\omega\)에 대하여, \(\omega_j:=\omega^j\)라 두자. 벡터 \((\omega_j^{k})_{0\leq k \leq n-1}\)는 순환 행렬 \(C_n\)의 고유벡터이며, 고유값은 \(\sum_{k=0}^{n-1}\omega_{j}^{k}a_k\)로 주어진다
예
- \(\omega=e^{2\pi i/4}=i\)
- 행렬 \(\mathcal{F}_4\)를 다음과 같이 정의하자
\[\mathcal{F}_4=\left( \begin{array}{cccc} 1 & 1 & 1 & 1 \\ 1 & \omega & \omega ^2 & \omega ^3 \\ 1 & \omega ^2 & 1 & \omega ^2 \\ 1 & \omega ^3 & \omega ^2 & \omega \\ \end{array} \right) \]
- 다음이 성립한다
\[ \mathcal{F}_4^{-1}C_4\mathcal{F}_4=\left( \begin{array}{cccc} a_0+a_1+a_2+a_3 & 0 & 0 & 0 \\ 0 & a_3 \omega ^3+a_2 \omega ^2+a_1 \omega +a_0 & 0 & 0 \\ 0 & 0 & a_1 \omega ^2+a_3 \omega ^2+a_0+a_2 & 0 \\ 0 & 0 & 0 & a_1 \omega ^3+a_2 \omega ^2+a_3 \omega +a_0 \\ \end{array} \right) \]
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
- http://mathoverflow.net/questions/104368/rational-solutions-to-x3-y3-z3-3xyz-1
- http://videolectures.net/mit18085f07_strang_lec23/
- Yilmaz, Nazmiye, Yasin Yazlik, and Necati Taskara. “On the Properties of Circulant Matrices Involving Generalized Tribonacci and Generalized Tribonacci-Lucas Numbers.” arXiv:1407.3925 [math], July 15, 2014. http://arxiv.org/abs/1407.3925.
관련된 항목들
수학용어번역
- circulant - 대한수학회 수학용어집
매스매티카 파일 및 계산 리소스
사전 형태의 자료
리뷰, 에세이, 강의노트
- Irwin Kra and Santiago R. Simanca On Circulant Matrices
관련논문
- Kushel, Olga, and Mikhail Tyaglov. “Circulants and Critical Points of Polynomials.” arXiv:1512.07983 [math], December 25, 2015. http://arxiv.org/abs/1512.07983.
메타데이터
위키데이터
- ID : Q245457
Spacy 패턴 목록
- [{'LOWER': 'circulant'}, {'LEMMA': 'matrix'}]