행렬의 크로네커 곱 (Kronecker product)

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

 

• http://www.google.com/search?hl=en&tbs=tl:1&q=
• 수학사 연표

 

 

메모

 

• Math Overflow http://mathoverflow.net/search?q=

 

 

 

 

매스매티카 파일 및 계산 리소스

• https://docs.google.com/leaf?id=0B8XXo8Tve1cxZGI2OGMwODEtN2E2OS00Njc1LTgwMjktMTQwNWU5OWUzZmYx&sort=name&layout=list&num=50

 

사전 형태의 자료

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Kronecker_product