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Mathematics-Online problems:

Interactive Problem 823: 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\;:\;5x_1^2+5x_2^2+8x_3^2-8x_1x_2+4x_1x_3+4x_2x_3-36=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$ . Fill the matrix $ T$ :

$ c_1:=\frac{1}{\sqrt{5}}$
$ c_2:=\frac{1}{3\sqrt{5}}$
$ c_3:=\frac{1}{3}$
$ T= \left(\rule{0pt}{8ex}\right.$
0 $ \cdot c_2$ $ \cdot c_3$
$ c_1$ $ \cdot c_2$ $ \cdot c_3$
$ \cdot c_1$ $ 2c_2$ - $ c_3$
$ \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 the normal form of $ Q$ is specified as:

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

The quadric $ Q$ is the following geometric figure:

n/a
ellipsoid
one-sheeted hyperboloid
hyperbolic paraboloid
elliptic cylinder
hyperbolic cylinder
cone

   

(Author: Christian Höfert)

see also:

  automatically generated: 8/11/2017