Mathematic-Online problems: Interactive Problem 70: Euclidian Normal Form of a Quadric

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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

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

symmetric!): $\displaystyle$

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

$\left.\rule{0pt}{8ex}\right)$ ,

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

$\left.\rule{0pt}{8ex}\right)$ ,

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

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

	0	0
0		0
0	0

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

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

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

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

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

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

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 is the following geometric figure:

n/a

ellipsoid

one-sheeted hyperboloid

hyperbolic paraboloid

zeppelin

cone

The signature of the quadric is:

n/a

equal to

the determinant of

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 and are orthogonal to each other. Thereby the point shall be element of $g_{1}$ . The intersection of $g_{1}$ and $g_{2}$ is: