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수학노트
(Dodgson condensation에서 넘어옴)
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개요

  • 행렬식을 계산하는 방법의 하나
  • nxn 행렬의 행렬식을 2x2 행렬의 행렬식을 반복적으로 계산하여 얻음

 

$n=3$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).