Mathematic-Online problems: Interactive Problem 293: Euclidian Normal Form of a Quadric

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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

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

in matrix form

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

with a symmetric matrix

and $x=(x_1, x_2, x_3)^{\rm {t}}$ .

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

b)

Give the Euclidian normal form of the quadric $Q_2: \, x^{\rm {t}}Ax=0$ and find an orthogonal matrix

, so that $T^{\,\rm {t}}AT$ is diagonal.
Sketch

with respect to normal form coordinates.

c)

Find the Euclidean normal form of

. Of what quadric type is

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 $T^{\,\rm {t}}AT= \left(\begin{array}{ccc} \lambda_1&0&0\\ 0&\lambda_2&0\\ 0&0&\lambda_3 \end{array}\right)$ .