"대칭곱 (symmetric power)과 대칭텐서"의 두 판 사이의 차이
Pythagoras0 (토론 | 기여) |
Pythagoras0 (토론 | 기여) |
||
| 5번째 줄: | 5번째 줄: | ||
==행렬의 대칭곱== | ==행렬의 대칭곱== | ||
| − | * $V$에 작용하는 선형변환 $A$를 생각하자 | + | * $V$에 작용하는 선형변환 $A=(a_{ij})$를 생각하자 |
| 20번째 줄: | 20번째 줄: | ||
1 & \left( | 1 & \left( | ||
\begin{array}{cc} | \begin{array}{cc} | ||
| − | + | a_{1,1} & a_{1,2} \\ | |
| − | + | a_{2,1} & a_{2,2} \\ | |
\end{array} | \end{array} | ||
\right) \\ | \right) \\ | ||
2 & \left( | 2 & \left( | ||
\begin{array}{ccc} | \begin{array}{ccc} | ||
| − | + | a_{1,1}^2 & a_{1,1} a_{1,2} & a_{1,2}^2 \\ | |
| − | 2 | + | 2 a_{1,1} a_{2,1} & a_{1,2} a_{2,1}+a_{1,1} a_{2,2} & 2 a_{1,2} a_{2,2} \\ |
| − | + | a_{2,1}^2 & a_{2,1} a_{2,2} & a_{2,2}^2 \\ | |
\end{array} | \end{array} | ||
\right) \\ | \right) \\ | ||
3 & \left( | 3 & \left( | ||
\begin{array}{cccc} | \begin{array}{cccc} | ||
| − | + | a_{1,1}^3 & a_{1,1}^2 a_{1,2} & a_{1,1} a_{1,2}^2 & a_{1,2}^3 \\ | |
| − | 3 | + | 3 a_{1,1}^2 a_{2,1} & a_{2,2} a_{1,1}^2+2 a_{1,2} a_{2,1} a_{1,1} & a_{2,1} a_{1,2}^2+2 a_{1,1} a_{2,2} a_{1,2} & 3 a_{1,2}^2 a_{2,2} \\ |
| − | 3 | + | 3 a_{1,1} a_{2,1}^2 & a_{1,2} a_{2,1}^2+2 a_{1,1} a_{2,2} a_{2,1} & a_{1,1} a_{2,2}^2+2 a_{1,2} a_{2,1} a_{2,2} & 3 a_{1,2} a_{2,2}^2 \\ |
| − | + | a_{2,1}^3 & a_{2,1}^2 a_{2,2} & a_{2,1} a_{2,2}^2 & a_{2,2}^3 \\ | |
\end{array} | \end{array} | ||
\right) \\ | \right) \\ | ||
4 & \left( | 4 & \left( | ||
\begin{array}{ccccc} | \begin{array}{ccccc} | ||
| − | + | a_{1,1}^4 & a_{1,1}^3 a_{1,2} & a_{1,1}^2 a_{1,2}^2 & a_{1,1} a_{1,2}^3 & a_{1,2}^4 \\ | |
| − | 4 | + | 4 a_{1,1}^3 a_{2,1} & a_{2,2} a_{1,1}^3+3 a_{1,2} a_{2,1} a_{1,1}^2 & 2 a_{1,2} a_{2,2} a_{1,1}^2+2 a_{1,2}^2 a_{2,1} a_{1,1} & a_{2,1} a_{1,2}^3+3 a_{1,1} a_{2,2} a_{1,2}^2 & 4 a_{1,2}^3 a_{2,2} \\ |
| − | 6 | + | 6 a_{1,1}^2 a_{2,1}^2 & 3 a_{2,1} a_{2,2} a_{1,1}^2+3 a_{1,2} a_{2,1}^2 a_{1,1} & a_{1,2}^2 a_{2,1}^2+4 a_{1,1} a_{1,2} a_{2,2} a_{2,1}+a_{1,1}^2 a_{2,2}^2 & 3 a_{2,1} a_{2,2} a_{1,2}^2+3 a_{1,1} a_{2,2}^2 a_{1,2} & 6 a_{1,2}^2 a_{2,2}^2 \\ |
| − | 4 | + | 4 a_{1,1} a_{2,1}^3 & a_{1,2} a_{2,1}^3+3 a_{1,1} a_{2,2} a_{2,1}^2 & 2 a_{1,2} a_{2,2} a_{2,1}^2+2 a_{1,1} a_{2,2}^2 a_{2,1} & a_{1,1} a_{2,2}^3+3 a_{1,2} a_{2,1} a_{2,2}^2 & 4 a_{1,2} a_{2,2}^3 \\ |
| − | + | a_{2,1}^4 & a_{2,1}^3 a_{2,2} & a_{2,1}^2 a_{2,2}^2 & a_{2,1} a_{2,2}^3 & a_{2,2}^4 \\ | |
| + | \end{array} | ||
| + | \right) \\ | ||
| + | \end{array} | ||
| + | $$ | ||
| + | |||
| + | ===$\dim V=3$인 경우=== | ||
| + | $$ | ||
| + | \begin{array}{c|c} | ||
| + | n & \operatorname{Sym}^nA \\ | ||
| + | \hline | ||
| + | 0 & \left( | ||
| + | \begin{array}{c} | ||
| + | 1 \\ | ||
| + | \end{array} | ||
| + | \right) \\ | ||
| + | 1 & \left( | ||
| + | \begin{array}{ccc} | ||
| + | a_{1,1} & a_{1,2} & a_{1,3} \\ | ||
| + | a_{2,1} & a_{2,2} & a_{2,3} \\ | ||
| + | a_{3,1} & a_{3,2} & a_{3,3} \\ | ||
| + | \end{array} | ||
| + | \right) \\ | ||
| + | 2 & \left( | ||
| + | \begin{array}{cccccc} | ||
| + | a_{1,1}^2 & a_{1,1} a_{1,2} & a_{1,1} a_{1,3} & a_{1,2}^2 & a_{1,2} a_{1,3} & a_{1,3}^2 \\ | ||
| + | 2 a_{1,1} a_{2,1} & a_{1,2} a_{2,1}+a_{1,1} a_{2,2} & a_{1,3} a_{2,1}+a_{1,1} a_{2,3} & 2 a_{1,2} a_{2,2} & a_{1,3} a_{2,2}+a_{1,2} a_{2,3} & 2 a_{1,3} a_{2,3} \\ | ||
| + | 2 a_{1,1} a_{3,1} & a_{1,2} a_{3,1}+a_{1,1} a_{3,2} & a_{1,3} a_{3,1}+a_{1,1} a_{3,3} & 2 a_{1,2} a_{3,2} & a_{1,3} a_{3,2}+a_{1,2} a_{3,3} & 2 a_{1,3} a_{3,3} \\ | ||
| + | a_{2,1}^2 & a_{2,1} a_{2,2} & a_{2,1} a_{2,3} & a_{2,2}^2 & a_{2,2} a_{2,3} & a_{2,3}^2 \\ | ||
| + | 2 a_{2,1} a_{3,1} & a_{2,2} a_{3,1}+a_{2,1} a_{3,2} & a_{2,3} a_{3,1}+a_{2,1} a_{3,3} & 2 a_{2,2} a_{3,2} & a_{2,3} a_{3,2}+a_{2,2} a_{3,3} & 2 a_{2,3} a_{3,3} \\ | ||
| + | a_{3,1}^2 & a_{3,1} a_{3,2} & a_{3,1} a_{3,3} & a_{3,2}^2 & a_{3,2} a_{3,3} & a_{3,3}^2 \\ | ||
\end{array} | \end{array} | ||
\right) \\ | \right) \\ | ||
2015년 5월 6일 (수) 20:14 판
개요
- 벡터공간 $V$에 대하여 대칭곱 $\operatorname{Sym}^n V$를 정의할 수 있다
- $V$에 작용하는 선형변환 $A$에 대하여 $\operatorname{Sym}^n A$를 정의할 수 있다
행렬의 대칭곱
- $V$에 작용하는 선형변환 $A=(a_{ij})$를 생각하자
$\dim V=2$인 경우
$$ \begin{array}{c|c} n & \operatorname{Sym}^nA \\ \hline 0 & \left( \begin{array}{c} 1 \\ \end{array} \right) \\ 1 & \left( \begin{array}{cc} a_{1,1} & a_{1,2} \\ a_{2,1} & a_{2,2} \\ \end{array} \right) \\ 2 & \left( \begin{array}{ccc} a_{1,1}^2 & a_{1,1} a_{1,2} & a_{1,2}^2 \\ 2 a_{1,1} a_{2,1} & a_{1,2} a_{2,1}+a_{1,1} a_{2,2} & 2 a_{1,2} a_{2,2} \\ a_{2,1}^2 & a_{2,1} a_{2,2} & a_{2,2}^2 \\ \end{array} \right) \\ 3 & \left( \begin{array}{cccc} a_{1,1}^3 & a_{1,1}^2 a_{1,2} & a_{1,1} a_{1,2}^2 & a_{1,2}^3 \\ 3 a_{1,1}^2 a_{2,1} & a_{2,2} a_{1,1}^2+2 a_{1,2} a_{2,1} a_{1,1} & a_{2,1} a_{1,2}^2+2 a_{1,1} a_{2,2} a_{1,2} & 3 a_{1,2}^2 a_{2,2} \\ 3 a_{1,1} a_{2,1}^2 & a_{1,2} a_{2,1}^2+2 a_{1,1} a_{2,2} a_{2,1} & a_{1,1} a_{2,2}^2+2 a_{1,2} a_{2,1} a_{2,2} & 3 a_{1,2} a_{2,2}^2 \\ a_{2,1}^3 & a_{2,1}^2 a_{2,2} & a_{2,1} a_{2,2}^2 & a_{2,2}^3 \\ \end{array} \right) \\ 4 & \left( \begin{array}{ccccc} a_{1,1}^4 & a_{1,1}^3 a_{1,2} & a_{1,1}^2 a_{1,2}^2 & a_{1,1} a_{1,2}^3 & a_{1,2}^4 \\ 4 a_{1,1}^3 a_{2,1} & a_{2,2} a_{1,1}^3+3 a_{1,2} a_{2,1} a_{1,1}^2 & 2 a_{1,2} a_{2,2} a_{1,1}^2+2 a_{1,2}^2 a_{2,1} a_{1,1} & a_{2,1} a_{1,2}^3+3 a_{1,1} a_{2,2} a_{1,2}^2 & 4 a_{1,2}^3 a_{2,2} \\ 6 a_{1,1}^2 a_{2,1}^2 & 3 a_{2,1} a_{2,2} a_{1,1}^2+3 a_{1,2} a_{2,1}^2 a_{1,1} & a_{1,2}^2 a_{2,1}^2+4 a_{1,1} a_{1,2} a_{2,2} a_{2,1}+a_{1,1}^2 a_{2,2}^2 & 3 a_{2,1} a_{2,2} a_{1,2}^2+3 a_{1,1} a_{2,2}^2 a_{1,2} & 6 a_{1,2}^2 a_{2,2}^2 \\ 4 a_{1,1} a_{2,1}^3 & a_{1,2} a_{2,1}^3+3 a_{1,1} a_{2,2} a_{2,1}^2 & 2 a_{1,2} a_{2,2} a_{2,1}^2+2 a_{1,1} a_{2,2}^2 a_{2,1} & a_{1,1} a_{2,2}^3+3 a_{1,2} a_{2,1} a_{2,2}^2 & 4 a_{1,2} a_{2,2}^3 \\ a_{2,1}^4 & a_{2,1}^3 a_{2,2} & a_{2,1}^2 a_{2,2}^2 & a_{2,1} a_{2,2}^3 & a_{2,2}^4 \\ \end{array} \right) \\ \end{array} $$
$\dim V=3$인 경우
$$ \begin{array}{c|c} n & \operatorname{Sym}^nA \\ \hline 0 & \left( \begin{array}{c} 1 \\ \end{array} \right) \\ 1 & \left( \begin{array}{ccc} a_{1,1} & a_{1,2} & a_{1,3} \\ a_{2,1} & a_{2,2} & a_{2,3} \\ a_{3,1} & a_{3,2} & a_{3,3} \\ \end{array} \right) \\ 2 & \left( \begin{array}{cccccc} a_{1,1}^2 & a_{1,1} a_{1,2} & a_{1,1} a_{1,3} & a_{1,2}^2 & a_{1,2} a_{1,3} & a_{1,3}^2 \\ 2 a_{1,1} a_{2,1} & a_{1,2} a_{2,1}+a_{1,1} a_{2,2} & a_{1,3} a_{2,1}+a_{1,1} a_{2,3} & 2 a_{1,2} a_{2,2} & a_{1,3} a_{2,2}+a_{1,2} a_{2,3} & 2 a_{1,3} a_{2,3} \\ 2 a_{1,1} a_{3,1} & a_{1,2} a_{3,1}+a_{1,1} a_{3,2} & a_{1,3} a_{3,1}+a_{1,1} a_{3,3} & 2 a_{1,2} a_{3,2} & a_{1,3} a_{3,2}+a_{1,2} a_{3,3} & 2 a_{1,3} a_{3,3} \\ a_{2,1}^2 & a_{2,1} a_{2,2} & a_{2,1} a_{2,3} & a_{2,2}^2 & a_{2,2} a_{2,3} & a_{2,3}^2 \\ 2 a_{2,1} a_{3,1} & a_{2,2} a_{3,1}+a_{2,1} a_{3,2} & a_{2,3} a_{3,1}+a_{2,1} a_{3,3} & 2 a_{2,2} a_{3,2} & a_{2,3} a_{3,2}+a_{2,2} a_{3,3} & 2 a_{2,3} a_{3,3} \\ a_{3,1}^2 & a_{3,1} a_{3,2} & a_{3,1} a_{3,3} & a_{3,2}^2 & a_{3,2} a_{3,3} & a_{3,3}^2 \\ \end{array} \right) \\ \end{array} $$
관련된 항목들
- 중복조합의 공식 H(n,r)=C(n+r-1,r)
- 외대수(exterior algebra)와 다중선형대수(multilinear algebra)
- 행렬의 크로네커 곱 (Kronecker product)
사전 형태의 자료
- http://en.wikipedia.org/wiki/Symmetric_algebra#Distinction_with_symmetric_tensors
- http://en.wikipedia.org/wiki/Symmetric_tensor
리뷰, 에세이, 강의노트