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Example: Principal Axis Transformation


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The quadric

$\displaystyle Q:\quad 8x_1^2 + 17x_2^2 + 20x_3^2 + 20x_1x_2 + 8x_1x_3 + 28x_2x_3 - 48x_1 -
114x_2 - 78x_3 + 207 =0
$

is to be brought to normal form.

In terms of matrices we have

$\displaystyle Q:\quad x^{\operatorname t}A x + 2b^{\operatorname t}x + c =0
$

where

\begin{displaymath}
A=\left(
\begin{array}{ccc}
8 & 10 & 4\\
10 & 17 & 14\\
4 ...
...ht)\,,\quad b=(-24,-57,-39)^{\operatorname t}\,,\quad c=207\,.
\end{displaymath}

Matrix $ A$ has the eigenvalues $ 36$, $ 9$ and 0, corresponding normalized eigenvectors are, for example,

$\displaystyle u_1=\frac{1}{3}(1,2,2)^{\operatorname t}\,,\quad
u_2=\frac{1}{3}(2,1,-2)^{\operatorname t}\,,\quad
u_3=\frac{1}{3}(2,-2,1)^{\operatorname t}\,.
$

After transformation $ x=Uy$ with $ U=(u_1,u_2,u_3)$ we obtain

$\displaystyle Q:\quad 36y_1^2 + 9 y_2^2 - 144y_1 - 18y_2 + 18y_3 + 207 = 0\,.
$

Completion of squares $ z_1=y_1-2$, $ z_2=y_2-1$, $ z_3=y_3$ yields

$\displaystyle Q:\quad 36z_1^2 + 9z_2^2 + 18z_3 + 54 = 0\,.
$

The translation $ w_1=z_1$, $ w_2=z_2$, $ w_3=z_3+3$ yields the normal form

$\displaystyle Q:\quad 36w_1^2 + 9w_2^2 + 18w_3 =0\,,
$

or after division by $ 9$

$\displaystyle Q:\quad 4w_1^2 + w_2^2 + 2w_3 =0\,.
$


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  automatisch erstellt am 13.  7. 2018