Ellipsoids, positive definite matrices (1): basic definitions and motivation

Antonio Sala, UPV

Difficulty: ** ,       Relevance: ,      Duration: 16:29

Summary:

This video presents motivation and basic definitions of: a sphere centered at the origin, generated by the quadratic form $x^{T}x\le 1$, an ellipsoid aligned with coordinate axes $x^{T}Dx\le 1$ ($D$ diagonal) and, last, generic ellipsoids $x^{T}Px\le 1$ ($P$ symmetric positive definite), introducing rotation matrices (orthogonal matrices) and diagonalization to explain the reasons leading to that last definition (the idea is discussed in more depth in the video [ellip2EN]).

Of course, 2D ellipses are a particular case of generic N-dimensional ellipsoids.

It presents Matlab examples with fimplicit, and motivates its use in least squares problems (geometry of spheres), weighted least squares (geometry of ellipses) and other control, robotics, statistics, material science, etc. contexts.

The video [ellip2EN], following this one, discusses in detail the interpretation of diagonalization, the ellipsoids not centered at the origin, and three equivalent ways to represent a given ellipsoid.

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