행렬의 크로네커 곱 (Kronecker product)

이 항목의 수학노트 원문주소

 

 

개요

 

 

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

 

 

 

관련된 항목들

 

 

매스매티카 파일 및 계산 리소스

 

 

 

수학용어번역

 

 

 

사전 형태의 자료

 

 

리뷰논문, 에세이, 강의노트

 

 

관련논문

 

 

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