행렬의 크로네커 곱 (Kronecker product)
이 항목의 수학노트 원문주소
==개요
- 두 행렬의 텐서곱 개념
- 두 벡터공간 V, W 를 정의역으로 하는 선형사상 A, B 에 대하여, \(V\otimes W\) 를 정의역으로 하는 선형사상 \(A\otimes B\) 을 정의할 수 있다
- \(A\otimes B\) 의 행렬표현으로부터 행렬의 크로네커 곱을 얻을 수 있다
==예
\(A=\left( \begin{array}{cc} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \end{array} \right)\)
\(B=\left( \begin{array}{ccc} b_{1,1} & b_{1,2} & b_{1,3} \\ b_{2,1} & b_{2,2} & b_{2,3} \\ b_{3,1} & b_{3,2} & b_{3,3} \end{array} \right)\)
\(A\otimes B=\left( \begin{array}{cccccc} a_{1,1} b_{1,1} & a_{1,1} b_{1,2} & a_{1,1} b_{1,3} & a_{1,2} b_{1,1} & a_{1,2} b_{1,2} & a_{1,2} b_{1,3} \\ a_{1,1} b_{2,1} & a_{1,1} b_{2,2} & a_{1,1} b_{2,3} & a_{1,2} b_{2,1} & a_{1,2} b_{2,2} & a_{1,2} b_{2,3} \\ a_{1,1} b_{3,1} & a_{1,1} b_{3,2} & a_{1,1} b_{3,3} & a_{1,2} b_{3,1} & a_{1,2} b_{3,2} & a_{1,2} b_{3,3} \\ a_{2,1} b_{1,1} & a_{2,1} b_{1,2} & a_{2,1} b_{1,3} & a_{2,2} b_{1,1} & a_{2,2} b_{1,2} & a_{2,2} b_{1,3} \\ a_{2,1} b_{2,1} & a_{2,1} b_{2,2} & a_{2,1} b_{2,3} & a_{2,2} b_{2,1} & a_{2,2} b_{2,2} & a_{2,2} b_{2,3} \\ a_{2,1} b_{3,1} & a_{2,1} b_{3,2} & a_{2,1} b_{3,3} & a_{2,2} b_{3,1} & a_{2,2} b_{3,2} & a_{2,2} b_{3,3} \end{array} \right)\)
\(v=\left( \begin{array}{c} v_1 \\ v_2 \end{array} \right)\)
\(w=\left( \begin{array}{c} w_1 \\ w_2 \\ w_3 \end{array} \right)\)
\(v\otimes w= \left( \begin{array}{c} v_1 w_1 \\ v_1 w_2 \\ v_1 w_3 \\ v_2 w_1 \\ v_2 w_2 \\ v_2 w_3 \end{array} \right)\)
\(Av \otimes Bw = (A\otimes B)( v\otimes w) =\left( \begin{array}{c} v_1 w_1 a_{1,1} b_{1,1}+v_2 w_1 a_{1,2} b_{1,1}+v_1 w_2 a_{1,1} b_{1,2}+v_2 w_2 a_{1,2} b_{1,2}+v_1 w_3 a_{1,1} b_{1,3}+v_2 w_3 a_{1,2} b_{1,3} \\ v_1 w_1 a_{1,1} b_{2,1}+v_2 w_1 a_{1,2} b_{2,1}+v_1 w_2 a_{1,1} b_{2,2}+v_2 w_2 a_{1,2} b_{2,2}+v_1 w_3 a_{1,1} b_{2,3}+v_2 w_3 a_{1,2} b_{2,3} \\ v_1 w_1 a_{1,1} b_{3,1}+v_2 w_1 a_{1,2} b_{3,1}+v_1 w_2 a_{1,1} b_{3,2}+v_2 w_2 a_{1,2} b_{3,2}+v_1 w_3 a_{1,1} b_{3,3}+v_2 w_3 a_{1,2} b_{3,3} \\ v_1 w_1 a_{2,1} b_{1,1}+v_2 w_1 a_{2,2} b_{1,1}+v_1 w_2 a_{2,1} b_{1,2}+v_2 w_2 a_{2,2} b_{1,2}+v_1 w_3 a_{2,1} b_{1,3}+v_2 w_3 a_{2,2} b_{1,3} \\ v_1 w_1 a_{2,1} b_{2,1}+v_2 w_1 a_{2,2} b_{2,1}+v_1 w_2 a_{2,1} b_{2,2}+v_2 w_2 a_{2,2} b_{2,2}+v_1 w_3 a_{2,1} b_{2,3}+v_2 w_3 a_{2,2} b_{2,3} \\ v_1 w_1 a_{2,1} b_{3,1}+v_2 w_1 a_{2,2} b_{3,1}+v_1 w_2 a_{2,1} b_{3,2}+v_2 w_2 a_{2,2} b_{3,2}+v_1 w_3 a_{2,1} b_{3,3}+v_2 w_3 a_{2,2} b_{3,3} \end{array} \right)\)
==역사
==메모
- Math Overflow http://mathoverflow.net/search?q=
==관련된 항목들
==매스매티카 파일 및 계산 리소스
- https://docs.google.com/leaf?id=0B8XXo8Tve1cxZGI2OGMwODEtN2E2OS00Njc1LTgwMjktMTQwNWU5OWUzZmYx&sort=name&layout=list&num=50
- http://www.wolframalpha.com/input/?i=
- http://functions.wolfram.com/
- NIST Digital Library of Mathematical Functions
- Abramowitz and Stegun Handbook of mathematical functions
- The On-Line Encyclopedia of Integer Sequences
- Numbers, constants and computation
- 매스매티카 파일 목록
수학용어번역
==사전 형태의 자료
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/Kronecker_product
- The Online Encyclopaedia of Mathematics
- NIST Digital Library of Mathematical Functions
- The World of Mathematical Equations
==리뷰논문, 에세이, 강의노트
==관련논문
==관련도서