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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 Problem Interactive Problem

The quadric in is defined by

a)
Give in matrix form

with a symmetric matrix and .

Show that is an eigenvector of corresponding to the eigenvalue . Find the other eigenvalues of .

b)
Give the Euclidian normal form of the quadric and find an orthogonal matrix , so that is diagonal.
Sketch with respect to normal form coordinates.
c)
Find the Euclidean normal form of . Of what quadric type is ?

a)

Characteristic polynomial: .

Eigenvalues:

b)
Eigenvectors:

corresponding to the eigenvalue : .

corresponding to the eigenvalue : .

Transformation matrix:
 2 3 6
with .

Normal form of : .

 n/a elliptic paraboloid hyperbolic paraboloid pair of intersecting planes ellipsoid

c)
After transformation of the linear term one gets:

.

After completion of squares one gets:

.