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Mathematics-Online problems:

Interactive Problem 70: Euclidian Normal Form of a Quadric


A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Given the quadric $ Q$ defined by

$\displaystyle Q\;:\;11x_{1}^{2}-2x_{2}^{2}+5x_{3}^{2}+12x_{1}x_{2}+12x_{2}x_{3}+
78x_{1}+32x_{2}+12x_3 + 133 =0 $

in $ \mathbb{R}^{3}$ .

Bring the quadric into matrix form

$\displaystyle x^{{\operatorname t}} Ax+2b^{{\operatorname t}}x+c=0$   (with $ x=(x_{1},x_{2},x_{3})^{{\operatorname t}}$ , and $ A$ symmetric!):$\displaystyle $

$ A= \left(\rule{0pt}{8ex}\right.$
$ \left.\rule{0pt}{8ex}\right)$ , $ b= \left(\rule{0pt}{8ex}\right.$
$ \left.\rule{0pt}{8ex}\right)$ , $ c=$ .  

Find the eigenvalues of $ A$ and give them in descending order into the diagonal of the following matrix $ D$ :

$ D= \left(\rule{0pt}{8ex}\right.$
0 0
0 0
0 0
$ \left.\rule{0pt}{8ex}\right)$ .

Find (using eigenvectors of $ A$ ) an orthogonal matrix $ T$ with $ T^{{\operatorname t}}AT=D$ . The first row of $ T$ shall consist of nonnegative entries. Bring the entries of $ T$ to the common denominator $ 7$ . Give the numerator values of $ T$ :

$ T= \frac{1}{7} \left(\rule{0pt}{8ex}\right.$
$ \left.\rule{0pt}{8ex}\right)$ .

The next intention is to find the Euclidean normal form of $ Q$ . First you have to find a representation of $ Q$ with respect to a coordinate system adapted to the symmetry (principle axes transformation), i.e. make the transformation $ x=Ty$ . In the new coordinates $ Q$ is specified as:

$ y_{1}^{2} + $ $ y_{2}^{2} + $ $ y_{3}^{2} + $ $ y_{1} + $ $ y_{2} + $ $ y_{3} + $ $ =0$ .

Eliminate the linear terms by translation: The transformation

$ y_{1}=z_{1}+ $ , $ \quad$ $ y_{2}=z_{2}+ $ , $ \quad$ $ y_{3}=z_{3}+ $

leads to the normal form

$ Q\;:\;$ $ z_{1}^{2} + $ $ z_{2}^{2} + $ $ z_{3}^{2}=$ .

The total transformation is:

$ x=Tz+ \left(\rule{0pt}{8ex}\right.$
$ \left.\rule{0pt}{8ex}\right)$ .

The quadric $ Q$ is the following geometric figure:

n/a
ellipsoid
one-sheeted hyperboloid
hyperbolic paraboloid
zeppelin
cone

The signature of the quadric $ Q$ is:

n/a
equal to $ 1$
the determinant of $ T$

The length of the principle axes of a quadric determines:

n/a
the intersection of the quadric and the coordinate axes
the intersection of the quadric and the unit sphere
the expansion of the quadric

Find two intersecting lines $ g_{1}$ and $ g_{2}$ , which are contained in $ Q$ and are orthogonal to each other. Thereby the point $ P(2,-9,-3)$ shall be element of $ g_{1}$ . The intersection $ S$ of $ g_{1}$ and $ g_{2}$ is:

$ S\Big($ , , $ \Big)$ .

Complete the following two normalized direction vectors of the lines $ g_{1}$ and $ g_{2}$ :

$ v_{1}= \frac{\sqrt{2}}{14} \left(\rule{0pt}{8ex}\right.$
$ 5$
$ \left.\rule{0pt}{8ex}\right)$ , $ v_{2}= \frac{\sqrt{2}}{14} \left(\rule{0pt}{8ex}\right.$
$ 1$
$ \left.\rule{0pt}{8ex}\right)$ .

   
(Authors: Hertweck/Höfert)

see also:


  automatically generated: 8/11/2017