"Symmetrizable generalized Cartan matrix"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| 2번째 줄: | 2번째 줄: | ||
* [[Generalized Cartan matrix]] | * [[Generalized Cartan matrix]] | ||
* symmetrizability condition the generalized Cartan matrix guarantees the existence of invariant bilinar forms | * symmetrizability condition the generalized Cartan matrix guarantees the existence of invariant bilinar forms | ||
| + | ;def | ||
| + | A [[generalized Cartan matrix]] $A$ is symmetrisable if there exists a non-singular diagonal matrix $D$ and a symmetric matrix $B$ such that $A=DB$. | ||
2015년 4월 2일 (목) 00:06 판
introduction
- Generalized Cartan matrix
- symmetrizability condition the generalized Cartan matrix guarantees the existence of invariant bilinar forms
- def
A generalized Cartan matrix $A$ is symmetrisable if there exists a non-singular diagonal matrix $D$ and a symmetric matrix $B$ such that $A=DB$.
example
- Cartan matrix of $G_2$
$$ A=\left( \begin{array}{cc} 2 & -1 \\ -3 & 2 \\ \end{array} \right) $$
- take $D$ as follows :
$$ D=\left( \begin{array}{cc} 3 & 0 \\ 0 & 1 \\ \end{array} \right) $$
- Then $DA=A^{t}D$ is a symmetric matrix
$$ \left( \begin{array}{cc} 6 & -3 \\ -3 & 2 \\ \end{array} \right) $$