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Ellipsoid/canonical form

Problem attached.

(a) Find the shortest and the largest distance from the origin to the surface of the ellipsoid.
(b) Find the principal axes of the ellipsoid.

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Given , consider the ellipsoid
(a) Find the shortest and largest distance from the origin to the surface of the ellipsoid.
(b) Find the principal axes of the ellipsoid.
(c) Find an orthogonal transformation such that , thereby reducing the equation to the canonical form.
Solution. (c) Since
, so we have
. Now we try to find an orthogonal matrix R such that R'AR is a diagonal matrix. How to do this? We first need to find the eigenvalues

of A. We can set up the following equation.

.
We get three roots which are three eigenvalues of A as follows. ...

Solution Summary

This shows how to find the longest and shortest distances from origin to an ellipsoid, the principal axes, and an orthogonal transformation. The canonical form is examined.

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