Ellipsoids (2): geometric interpretation of diagonalization, alternate representations of ellipsoids

Antonio Sala, UPV

Difficulty: ** ,       Relevance: PIC,      Duration: 14:59

Materials:    [ EllipsIntroEN.zip]

Summary:

This video is a continuation of [ellip1EN]. Here, we insist on the meaning of the diagonalization of a defined positive matrix, to determine semiaxes and their directions (recovering the diagonal scalings and rotations that generate it).

The second part of the video discusses alternative ways of representing ellipsoids: as a quadratic form xT Px 1, as xT Q1x 1, or as the image of the unit sphere by a matrix L (linear map). It also presents the ellipsoid not centered at the origin.

The video [ellip3EN], a continuation of this one, presents 3D examples and degenerate cases of non-invertible P or Q.

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

© 2024, A. Sala. All rights reserved for materials from authors affiliated to Universitat Politecnica de Valencia.
Please consult original source/authors for info regarding rights of materials from third parties.