Ellipsoids (5): inclusion, inscribed/circunscribed sphere

Antonio Sala, UPV

Difficulty: **** ,       Relevance: ,      Duration: 18:03

Summary:

This video develops the conditions for an ellipsoid to be included into another. Basically, when written in direct form, $x^T{P}_{1}x\le 1$ is included in $x^T{P}_{2}x\le 1$ if and only if $P_1-P_2\succcurlyeq 0$. Conversely, in inverted form, $x^T{Q}_1^{-1}x\le 1$ is included in $x^T{Q}_2^{-1}x\le 1$ if and only if $Q_2-Q_1\succcurlyeq 0$.

The video dedicates two thirds of its duration to the proof of the above conditions. As a particular case, if one of the two ellipsoids is a sphere of radius $\rho$, that is, $x^T{\rho }^{-2}x\le 1$, the application of the previous conditions results in conditions on the eigenvalues of $P$ or $Q$ that give the minimum radius and maximum radius of the ellipsoid (radius of the inscribed and circumscribed spheres, respectively).

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