Ellipsoids (4): equivalence of representations

Antonio Sala, UPV

Difficulty: **** ,       Relevance: ,      Duration: 14:25

Summary:

This video presents a discussion on the equivalence of representations of ellipsoids: (1) direct $x^{T}Px\le 1$, (2) inverse $x^T{Q}^{-1}x\le 1$ and (3) as a linear transformation of a sphere $x=Lu$, $u^{T}u\le 1$.

This video could be considered optional if you’re just getting started with all of this, as it discusses refinements that might not be necessary for a first approach to these topics.

The topics discussed here complete the ideas presented in the video [ellip2EN], to be watched prior to this one. Specifically, we discuss how to transform representation (3) to representation (2) when $L$ is not invertible, either because it has excess columns or excess rows.

Then, the final part of the video discusses the transformation from representation (2) to (3) by factoring $Q=L{L}^T$ either by Choleski’s method ($L$ triangular) or by diagonalization ($L=V\sqrt{D}{V}^T$ symmetric from the diagonalization of $Q=VD{V}^T$). This $L$ is not unique because we have one “rotation” degree of freedom as sphere $u^{T}u\le 1$ renders the same sphere when rotated, so many $L$ representing the same ellipsoid exist (square root of a matrix is not unique).

*Link to my [ whole collection] of videos in English. Link to larger [ Colección completa] in Spanish.