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Principal Axis Transformation


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By coordinate transformation (rotation and translation) a quadric in $ \mathbb{R}^n$ can be brought to normal form:

$\displaystyle x^{\operatorname t}A x + 2 b^{\operatorname t}x + c =
\sum_{i=1}^m \lambda_i w_i^2 +
2\beta w_{m+1} + \gamma
\,,
$

where $ \beta\gamma=0$.

\includegraphics[width=\moimagesize]{a_hauptachsentrafo}

The columns of the rotation matrix $ U$ contain the eigenvectors $ u_i$ of $ A$ the directions of which are called principal axes.

Example:


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  automatically generated 7/13/2018