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Mathematics-Online problems:

Interactive Problem 319: Euclidean 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 matrix

$\displaystyle A = \left ( \begin{array}{ccc} -1 & 1 & -\sqrt{2} \\ 1 & -1 & -\sqrt{2} \\ -\sqrt{2} & -\sqrt{2} & 0 \end{array} \right ) . $

a)
Find the eigenvalues and the eigenvectors of $ A .$
For control: 2 is eigenvalue of $ A .$

b)
Find an orthonormal basis of $ \mathbb{R}^3 ,$ consisting of eigenvectors of $ A$ .

c)
What's the Euclidean normal form of the quadric

$\displaystyle Q_0: x^{\mathrm{t}}Ax = 0 \ ? $

Give the corresponding transformation

$\displaystyle x = T \hat{x}\ . $

d)
Which $ c \in \mathbb{R}$ make the quadric

$\displaystyle Q_c: x^{\mathrm{t}}Ax + c = 0 $

to contain a line?

Answer:

a)
Characteristic polynomial:          $ \lambda^3+$ $ \lambda^2+$ $ \lambda+$ .

Eigenvalues:         $ \lambda_{1/2}=$          $ \lambda_3=$ .

Eigenvectors:

corresponding to $ \lambda_{1/2}$ :
$ \left(\rule{0pt}{8ex}\right.$
$ \sqrt{2}$
0
$ \left.\rule{0pt}{8ex}\right)\qquad \left(\rule{0pt}{8ex}\right.$
$ \sqrt{2}$
$ \left.\rule{0pt}{8ex}\right)$
corresponding to $ \lambda_3$ :
$ \left(\rule{0pt}{8ex}\right.$
$ -\sqrt{2}$
$ \left.\rule{0pt}{8ex}\right)$

b)
Transformation matrix $ T$ with

$\displaystyle T^{\operatorname t}AT= \left(\begin{array}{ccc} \lambda_3 & 0 & 0\\ 0 & \lambda_{1/2} & 0\\ 0 & 0 & \lambda_{1/2} \end{array}\right).$

$ T=\dfrac{1}{6}\left(\rule{0pt}{8ex}\right.$
    $ \sqrt{6}$      $ \sqrt{3}$
0 $ \sqrt{3}$
$ -3\sqrt{2}$     $ \sqrt{3}$      $ \sqrt{6}$
$ \left.\rule{0pt}{8ex}\right)$
Normal form:          $ \hat{x}_1^2+$ $ \hat{x}_2^2+$ $ \hat{x}_3^2=0$ .

c)
n/a
$ c<0$
$ c>0$
$ c\le0$
$ c\ge0$
$ c=0$
$ c\ne0$
$ c \in \mathbb{R}$

   
(Authors: Knödler/Höfert)

see also:

  automatically generated: 8/11/2017