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Mathematics-Online problems:

Interactive Problem 293: 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

The quadric $ Q_1$ in $ \mathbb{R}^3$ is defined by

$\displaystyle Q_1: \, x_1^2+20x_3^2+12x_1x_2+12x_1x_3+24x_2x_3-14x_2+7x_3-7=0. $


a)
Give $ Q_1$ in matrix form

$\displaystyle Q_1: \, x^{\rm {t}}Ax+2a^{\rm {t}}x+c = 0, $

with a symmetric matrix $ A$ and $ x=(x_1, x_2, x_3)^{\rm {t}}$ .

Show that $ v_1=(2, 3, 6)^{\rm {t}}$ is an eigenvector of $ A$ corresponding to the eigenvalue $ \lambda_1$ . Find the other eigenvalues of $ A$ .

b)
Give the Euclidian normal form of the quadric $ Q_2: \,
x^{\rm {t}}Ax=0$ and find an orthogonal matrix $ T$ , so that $ T^{\,\rm {t}}AT$ is diagonal.
Sketch $ Q_2$ with respect to normal form coordinates.
c)
Find the Euclidean normal form of $ Q_1$ . Of what quadric type is $ Q_1$ ?

Answer:

a)
$ Av_1=\lambda_1v_1\qquad\lambda_1=$

Characteristic polynomial: $ \chi_A(\lambda)=$ $ \lambda^3+$ $ \lambda^2+$ $ \lambda+$ .

Eigenvalues: $ \lambda_2<\lambda_3$          $ \lambda_2=$          $ \lambda_3=$

b)
Eigenvectors:

corresponding to the eigenvalue $ \lambda_2$ : $ v_2=\Big(3,$ $ ,$ $ \Big)^{\operatorname t}$ .

corresponding to the eigenvalue $ \lambda_3$ : $ v_3=\Big($ $ ,$ $ ,3\Big)^{\operatorname t}$ .

Transformation matrix:
$ T=1\Big/$ $ \left(\rule{0pt}{8ex}\right.$
2
3
6
$ \left.\rule{0pt}{8ex}\right)$ with $ T^{\,\rm {t}}AT= \left(\begin{array}{ccc}
\lambda_1&0&0\\
0&\lambda_2&0\\
0&0&\lambda_3
\end{array}\right)$ .

Normal form of $ Q_2$ : $ y_1^2+$ $ y_2^2+$ $ y_3^2=0$ .

The quadirc $ Q_2$ is a
n/a
elliptic paraboloid
hyperbolic paraboloid
pair of intersecting planes
ellipsoid

c)
After transformation of the linear term one gets:

$ Q_1:$ $ y_1^2+$ $ y_2^2+$ $ y_3^2+$ $ y_1+$ $ y_2+$ $ y_3+$ $ =0$ .

After completion of squares one gets:

$ Q_1:$ $ z_1^2+$ $ z_2^2+$ $ z_3^2+$ $ z_1+$ $ z_2+$ $ z_3+$ $ =0$ .

The quadric $ Q_1$ is a
n/a
elliptic paraboloid
hyperbolic paraboloid
pair of intersecting planes
ellipsoid


   
(Authors: Knödler/Höfert)

see also:


  automatically generated: 8/11/2017