"도지슨 응축"의 두 판 사이의 차이
Pythagoras0 (토론 | 기여) |
Pythagoras0 (토론 | 기여) |
||
| 14번째 줄: | 14번째 줄: | ||
v & w & x | v & w & x | ||
\end{vmatrix} | \end{vmatrix} | ||
| − | \ | + | = |
| + | \begin{vmatrix} | ||
| + | \begin{vmatrix} | ||
| + | p & q \\ | ||
| + | s & t | ||
| + | \end{vmatrix} & | ||
| + | \begin{vmatrix} | ||
| + | q & r \\ | ||
| + | t & u | ||
| + | \end{vmatrix} \\ | ||
| + | \begin{vmatrix} | ||
| + | s & t \\ | ||
| + | v & w | ||
| + | \end{vmatrix} & | ||
| + | \begin{vmatrix} | ||
| + | t & u \\ | ||
| + | w & x | ||
| + | \end{vmatrix} | ||
| + | \end{vmatrix} | ||
| + | = | ||
\begin{vmatrix} | \begin{vmatrix} | ||
-q s+p t & -r t+q u \\ | -q s+p t & -r t+q u \\ | ||
-t v+s w & -u w+t x | -t v+s w & -u w+t x | ||
\end{vmatrix} | \end{vmatrix} | ||
| − | + | = -r t v+q u v+r s w-p u w-q s x+p t x | |
| + | $$ | ||
| + | |||
| + | $$ | ||
| + | \begin{vmatrix} | ||
| + | 5 & 1 & 2 \\ | ||
| + | 6 & 1 & 3 \\ | ||
| + | 7 & 5 & 4 \\ | ||
| + | \end{vmatrix}= | ||
| + | \begin{vmatrix} | ||
| + | -1 & 1 \\ | ||
| + | 23 & -11 \\ | ||
| + | \end{vmatrix}=-12 | ||
$$ | $$ | ||
| + | ===$n=4$인 경우=== | ||
| + | $$ | ||
| + | \begin{vmatrix} | ||
| + | 2 & 1 & -1 & -3 \\ | ||
| + | 1 & -2 & 3 & 0 \\ | ||
| + | 3 & 1 & 2 & -1 \\ | ||
| + | 0 & -2 & 3 & 1 \\ | ||
| + | \end{vmatrix}= | ||
| + | \begin{vmatrix} | ||
| + | -5 & 1 & 9 \\ | ||
| + | 7 & -7 & -3 \\ | ||
| + | -6 & 7 & 5 \\ | ||
| + | \end{vmatrix}= | ||
| + | \begin{vmatrix} | ||
| + | -14 & 20 \\ | ||
| + | 7 & -7 \\ | ||
| + | \end{vmatrix} | ||
| + | =6 | ||
| + | $$ | ||
| + | |||
==역사== | ==역사== | ||
2013년 11월 22일 (금) 08:30 판
개요
- 행렬식을 계산하는 방법의 하나
- nxn 행렬의 행렬식을 2x2 행렬의 행렬식을 반복적으로 계산하여 얻음
예
$n=3$의 경우
$$ \begin{vmatrix} p & q & r \\ s & t & u \\ v & w & x \end{vmatrix} = \begin{vmatrix} \begin{vmatrix} p & q \\ s & t \end{vmatrix} & \begin{vmatrix} q & r \\ t & u \end{vmatrix} \\ \begin{vmatrix} s & t \\ v & w \end{vmatrix} & \begin{vmatrix} t & u \\ w & x \end{vmatrix} \end{vmatrix} = \begin{vmatrix} -q s+p t & -r t+q u \\ -t v+s w & -u w+t x \end{vmatrix} = -r t v+q u v+r s w-p u w-q s x+p t x $$
$$ \begin{vmatrix} 5 & 1 & 2 \\ 6 & 1 & 3 \\ 7 & 5 & 4 \\ \end{vmatrix}= \begin{vmatrix} -1 & 1 \\ 23 & -11 \\ \end{vmatrix}=-12 $$
$n=4$인 경우
$$ \begin{vmatrix} 2 & 1 & -1 & -3 \\ 1 & -2 & 3 & 0 \\ 3 & 1 & 2 & -1 \\ 0 & -2 & 3 & 1 \\ \end{vmatrix}= \begin{vmatrix} -5 & 1 & 9 \\ 7 & -7 & -3 \\ -6 & 7 & 5 \\ \end{vmatrix}= \begin{vmatrix} -14 & 20 \\ 7 & -7 \\ \end{vmatrix} =6 $$
역사
메모
- 1986 Robbins-Rumsey lambda determinant
- Dodgson’s condensation method for computing determinants has led to the notion of alternating sign matrices and to their remarkable combinatorics. These topics have connections with the 6-vertex model in physics and statistical mechanics and with much recent work on graphical condensation, group characters, and a whole lot more.
관련된 항목들
매스매티카 파일 및 계산 리소스
수학용어번역
사전 형태의 자료
리뷰논문, 에세이, 강의노트
- Hone, Andrew N. W. 2006. “Dodgson Condensation, Alternating Signs and Square Ice.” Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences 364 (1849): 3183–3198. doi:10.1098/rsta.2006.1887.
- Abeles, Francine F. 2008. “Dodgson Condensation: The Historical and Mathematical Development of an Experimental Method.” Linear Algebra and Its Applications 429 (2-3): 429–438. doi:10.1016/j.laa.2007.11.022.
- Lewis Carroll and His Telescoping Determinants
관련논문
- Berliner, Adam, and Richard A. Brualdi. 2008. “A Combinatorial Proof of the Dodgson/Muir Determinantal Identity.” International Journal of Information & Systems Sciences 4 (1): 1–7.
- Zeilberger, Doron. 1997. “Dodgson’s Determinant-Evaluation Rule Proved by Two-Timing Men and Women.” Electronic Journal of Combinatorics 4 (2): Research Paper 22, approx. 2 pp. (electronic).