"코쉬 행렬과 행렬식"의 두 판 사이의 차이

2021년 2월 17일 (수) 06:01 기준 최신판

개요

행렬 \(A=({\frac{1}{x_i-y_j}})_{1\le i,j\le n}\)를 크기 n인 코쉬 행렬이라 함
행렬식

\[ \det \left(\frac{1}{x _i-y _j}\right) _{1\le i,j \le n}=(-1)^{\binom{n}{2}}\frac{\prod _{1\le i < j\le n} (x_j-x _i)(y _j-y _i)}{\prod _{i,j=1}^n (x _i-y _j)} \] \[ \det \left(\frac{1}{x _i+y _j}\right) _{1\le i,j \le n}=\frac{\prod _{1\le i < j\le n} (x_j-x _i)(y _j-y _i)}{\prod _{i,j=1}^n (x _i+y _j)} \]

예

n=1인 경우

\(\left( \begin{array}{c} \frac{1}{x_1-y_1} \end{array} \right)\)

n=2인 경우

코쉬 행렬

\[\left( \begin{array}{cc} \frac{1}{x_1-y_1} & \frac{1}{x_1-y_2} \\ \frac{1}{x_2-y_1} & \frac{1}{x_2-y_2} \end{array} \right)\]

행렬식

\[ \frac{\left(x_1-x_2\right) \left(y_1-y_2\right)}{\left(x_1-y_1\right) \left(y_1-x_2\right) \left(x_1-y_2\right) \left(x_2-y_2\right)} \]

n=3인 경우

코쉬 행렬은

\[\left( \begin{array}{ccc} \frac{1}{x_1-y_1} & \frac{1}{x_1-y_2} & \frac{1}{x_1-y_3} \\ \frac{1}{x_2-y_1} & \frac{1}{x_2-y_2} & \frac{1}{x_2-y_3} \\ \frac{1}{x_3-y_1} & \frac{1}{x_3-y_2} & \frac{1}{x_3-y_3} \end{array} \right)\]

행렬식은

\[-\frac{\left(-x_1+x_2\right) \left(-x_1+x_3\right) \left(-x_2+x_3\right) \left(y_1-y_2\right) \left(y_1-y_3\right) \left(y_2-y_3\right)}{\left(x_3-y_1\right) \left(-x_1+y_1\right) \left(-x_2+y_1\right) \left(x_2-y_2\right) \left(x_3-y_2\right) \left(-x_1+y_2\right) \left(x_1-y_3\right) \left(x_2-y_3\right) \left(x_3-y_3\right)}\]

n=4인 경우

코쉬 행렬은

\[\left( \begin{array}{cccc} \frac{1}{x_1-y_1} & \frac{1}{x_1-y_2} & \frac{1}{x_1-y_3} & \frac{1}{x_1-y_4} \\ \frac{1}{x_2-y_1} & \frac{1}{x_2-y_2} & \frac{1}{x_2-y_3} & \frac{1}{x_2-y_4} \\ \frac{1}{x_3-y_1} & \frac{1}{x_3-y_2} & \frac{1}{x_3-y_3} & \frac{1}{x_3-y_4} \\ \frac{1}{x_4-y_1} & \frac{1}{x_4-y_2} & \frac{1}{x_4-y_3} & \frac{1}{x_4-y_4} \end{array} \right)\]

역사

수학사 연표

메모

http://mathoverflow.net/questions/20609/what-role-does-cauchys-determinant-identity-play-in-combinatorics

매스매티카 파일 및 계산 리소스

https://docs.google.com/leaf?id=0B8XXo8Tve1cxM2E1ODYzMGUtYTJhMi00MmYxLWEzZDMtZDI2NmZmMWZmMDdm&sort=name&layout=list&num=50

사전 형태의 자료

http://en.wikipedia.org/wiki/Cauchy_matrix

메타데이터

위키데이터

ID : Q2915997

Spacy 패턴 목록

[{'LOWER': 'cauchy'}, {'LEMMA': 'matrix'}]

@@ 1번째 줄: / 1번째 줄: @@
-==이 항목의 스프링노트 원문주소==
-* [[코쉬 행렬과 행렬식]]<br>
 ==개요==
+* 행렬 <math>A=({\frac{1}{x_i-y_j}})_{1\le i,j\le n}</math>를 크기 n인 코쉬 행렬이라 함
+* 행렬식
+:<math>
+\det \left(\frac{1}{x _i-y _j}\right) _{1\le i,j \le n}=(-1)^{\binom{n}{2}}\frac{\prod _{1\le i < j\le n} (x_j-x _i)(y _j-y _i)}{\prod _{i,j=1}^n (x _i-y _j)}
+</math>
+:<math>
+\det \left(\frac{1}{x _i+y _j}\right) _{1\le i,j \le n}=\frac{\prod _{1\le i < j\le n} (x_j-x _i)(y _j-y _i)}{\prod _{i,j=1}^n (x _i+y _j)}
+</math>
-<math>a_{ij}={\frac{1}{x_i-y_j}}</math>
-<math>\left( \begin{array}{c}  \frac{1}{x_1-y_1} \end{array} \right)</math>
+==예==
-<math>\left( \begin{array}{cc}  \frac{1}{x_1-y_1} & \frac{1}{x_1-y_2} \\  \frac{1}{x_2-y_1} & \frac{1}{x_2-y_2} \end{array} \right)</math>
+===n=1인 경우===
+* <math>\left( \begin{array}{c}  \frac{1}{x_1-y_1} \end{array} \right)</math>
-==n=3인 경우==
+===n=2인 경우===
+* 코쉬 행렬
+:<math>\left( \begin{array}{cc}  \frac{1}{x_1-y_1} & \frac{1}{x_1-y_2} \\  \frac{1}{x_2-y_1} & \frac{1}{x_2-y_2} \end{array} \right)</math>
+* 행렬식
+:<math>
+\frac{\left(x_1-x_2\right) \left(y_1-y_2\right)}{\left(x_1-y_1\right) \left(y_1-x_2\right) \left(x_1-y_2\right) \left(x_2-y_2\right)}
+</math>
-<math>\left( \begin{array}{ccc}  \frac{1}{x_1-y_1} & \frac{1}{x_1-y_2} & \frac{1}{x_1-y_3} \\  \frac{1}{x_2-y_1} & \frac{1}{x_2-y_2} & \frac{1}{x_2-y_3} \\  \frac{1}{x_3-y_1} & \frac{1}{x_3-y_2} & \frac{1}{x_3-y_3} \end{array} \right)</math>
+===n=3인 경우===
+* 코쉬 행렬은
+:<math>\left( \begin{array}{ccc}  \frac{1}{x_1-y_1} & \frac{1}{x_1-y_2} & \frac{1}{x_1-y_3} \\  \frac{1}{x_2-y_1} & \frac{1}{x_2-y_2} & \frac{1}{x_2-y_3} \\  \frac{1}{x_3-y_1} & \frac{1}{x_3-y_2} & \frac{1}{x_3-y_3} \end{array} \right)</math>
+* 행렬식은
+:<math>-\frac{\left(-x_1+x_2\right) \left(-x_1+x_3\right) \left(-x_2+x_3\right) \left(y_1-y_2\right) \left(y_1-y_3\right) \left(y_2-y_3\right)}{\left(x_3-y_1\right) \left(-x_1+y_1\right) \left(-x_2+y_1\right) \left(x_2-y_2\right) \left(x_3-y_2\right) \left(-x_1+y_2\right) \left(x_1-y_3\right) \left(x_2-y_3\right) \left(x_3-y_3\right)}</math>
-행렬식은
-<math>-\frac{\left(-x_1+x_2\right) \left(-x_1+x_3\right) \left(-x_2+x_3\right) \left(y_1-y_2\right) \left(y_1-y_3\right) \left(y_2-y_3\right)}{\left(x_3-y_1\right) \left(-x_1+y_1\right) \left(-x_2+y_1\right) \left(x_2-y_2\right) \left(x_3-y_2\right) \left(-x_1+y_2\right) \left(x_1-y_3\right) \left(x_2-y_3\right) \left(x_3-y_3\right)}</math>
+===n=4인 경우===
+* 코쉬 행렬은
+:<math>\left( \begin{array}{cccc}  \frac{1}{x_1-y_1} & \frac{1}{x_1-y_2} & \frac{1}{x_1-y_3} & \frac{1}{x_1-y_4} \\  \frac{1}{x_2-y_1} & \frac{1}{x_2-y_2} & \frac{1}{x_2-y_3} & \frac{1}{x_2-y_4} \\  \frac{1}{x_3-y_1} & \frac{1}{x_3-y_2} & \frac{1}{x_3-y_3} & \frac{1}{x_3-y_4} \\  \frac{1}{x_4-y_1} & \frac{1}{x_4-y_2} & \frac{1}{x_4-y_3} & \frac{1}{x_4-y_4} \end{array} \right)</math>
-n=4인 경우
-<math>\left( \begin{array}{cccc}  \frac{1}{x_1-y_1} & \frac{1}{x_1-y_2} & \frac{1}{x_1-y_3} & \frac{1}{x_1-y_4} \\  \frac{1}{x_2-y_1} & \frac{1}{x_2-y_2} & \frac{1}{x_2-y_3} & \frac{1}{x_2-y_4} \\  \frac{1}{x_3-y_1} & \frac{1}{x_3-y_2} & \frac{1}{x_3-y_3} & \frac{1}{x_3-y_4} \\  \frac{1}{x_4-y_1} & \frac{1}{x_4-y_2} & \frac{1}{x_4-y_3} & \frac{1}{x_4-y_4} \end{array} \right)</math>
 ==역사==
+* [[수학사 연표]]
-* http://www.google.com/search?hl=en&tbs=tl:1&q=
-* [[수학사연표 (역사)|수학사연표]]
-*
 ==메모==
+* http://mathoverflow.net/questions/20609/what-role-does-cauchys-determinant-identity-play-in-combinatorics
 ==관련된 항목들==
+* [[반데몬드 행렬과 행렬식 (Vandermonde matrix)]]
+* [[힐버트 행렬]]
-* [[힐버트 행렬]]<br>
-==수학용어번역==
-* 단어사전 http://www.google.com/dictionary?langpair=en|ko&q=
-* 발음사전 http://www.forvo.com/search/
-* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]<br>
-** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
-* [http://www.nktech.net/science/term/term_l.jsp?l_mode=cate&s_code_cd=MA 남·북한수학용어비교]
-* [http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 대한수학회 수학용어한글화 게시판]
 ==매스매티카 파일 및 계산 리소스==
 * https://docs.google.com/leaf?id=0B8XXo8Tve1cxM2E1ODYzMGUtYTJhMi00MmYxLWEzZDMtZDI2NmZmMWZmMDdm&sort=name&layout=list&num=50
-* http://www.wolframalpha.com/input/?i=
-* http://functions.wolfram.com/
-* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
-* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
-* [http://numbers.computation.free.fr/Constants/constants.html Numbers, constants and computation]
-* [[매스매티카 파일 목록]]
-==사전 형태의 자료==
+==사전 형태의 자료==
 * http://en.wikipedia.org/wiki/Cauchy_matrix
-* http://en.wikipedia.org/wiki/
-* http://www.wolframalpha.com/input/?i=
-* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
-* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]<br>
-** http://www.research.att.com/~njas/sequences/?q=
+==관련논문==
+* Ishikawa, Masao, Soichi Okada, Hiroyuki Tagawa, and Jiang Zeng. “Generalizations of Cauchy’s Determinant and Schur’s Pfaffian.” Advances in Applied Mathematics 36, no. 3 (2006): 251–87. doi:10.1016/j.aam.2005.07.001.
+* Chen, William Y. C., Christian Krattenthaler, and Arthur L. B. Yang. “The Flagged Cauchy Determinant.” Graphs and Combinatorics 21, no. 1 (2005): 51–62. doi:10.1007/s00373-004-0593-9.
-==블로그==
+[[분류:선형대수학]]
+[[분류:대칭다항식]]
+[[분류:행렬식]]
-네이버 ]
+==메타데이터==
+===위키데이터===
+* ID :  [https://www.wikidata.org/wiki/Q2915997 Q2915997]
+===Spacy 패턴 목록===
+* [{'LOWER': 'cauchy'}, {'LEMMA': 'matrix'}]