Mathematik-Online lexicon: Example: Simplistic Classification of Quadrics

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	Mathematik-Online lexicon:
	Example: Simplistic Classification of Quadrics

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overview

Given the quadric

$\displaystyle Q:\quad x_1^2 +\lambda x_2^2 +2x_2 +\lambda =0$

with $\lambda\in\mathbb{R}$ , or in terms of matrices

$\displaystyle Q:\quad x^{\operatorname t}A x + 2b^{\operatorname t}x + c =0$

where

$\begin{displaymath} A=\left( \begin{array}{cc} 1 & 0 \\ 0 & \lambda\\ \end{arr... ...0 & 1\\ 0 & 1 & 0\\ 1 & 0 & \lambda\\ \end{array}\right)\,. \end{displaymath}$

For $\lambda=0$ we have $\operatorname{Rg}(A)=1$ . Otherwise: $\operatorname{Rg}(A)=2$ . For $\lambda=\pm 1$ we have $\operatorname{Rg}(\tilde{A})=2$ , otherwise: $\operatorname{Rg}(\tilde{A})=3$ .

Thus, is parabolic for $\lambda=0$ , conic for $\lambda=\pm 1$ and a central quadric for all other $\lambda$ .

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automatisch erstellt am 13. 7. 2018