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Ellipsoid/canonical form

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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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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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Please see the attachment.

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 provided by:
Changping Wang, MA
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
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  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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