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Ellipsoids – Quadratic Forms

Posts on the math of ellipsoids defined by quadratic forms

Orthogonal projections of multidimensional ellipsoids – V – relation between the inverse matrices of the involved quadratic forms and of respective covariance matrices

This series started with four main questions. The first two were: How do we know that an orthogonal projection of a (n-1)-dimensional ellipsoid from its n-dimensional vector space onto a lower p-dimensional sub-space leads to yet another ellipsoid? And how can we derive the matrix defining the quadratic form of the resulting lower dimensional ellipsoid from the matrix describing the… Read More »Orthogonal projections of multidimensional ellipsoids – V – relation between the inverse matrices of the involved quadratic forms and of respective covariance matrices

Orthogonal projections of multidimensional ellipsoids – IV – the relation to a Schur complement of the quadratic form matrix

The first three posts in this series showed that the orthogonal projection of a (n-1)-dimensional ellipsoid from a n-dimensional Euclidean space to a (n-1)-dimensional subspace has a surface which is a (n-2)-dimensional ellipsoid. In this fourth post we will extend our insights to projections down onto a p-dimensional sub-spaces with 1 ≤ p < n. The sub-space has a (n-p)-dimensional… Read More »Orthogonal projections of multidimensional ellipsoids – IV – the relation to a Schur complement of the quadratic form matrix

Orthogonal projections of multidimensional ellipsoids – III – arbitrary vectors for the projection direction and cuts of the unit sphere

The first two posts of this series showed that the orthogonal projection of an ellipsoid in the ℝn onto a (n-1) dimensional sub-space has a (n-2)-dimensional ellipsoidal surface. The tangential points on the original ellipsoid which control the surface of the projection could be derived by linear transformation A of well defined, but special vectors of the (n-1)-dimensional unit sphere.… Read More »Orthogonal projections of multidimensional ellipsoids – III – arbitrary vectors for the projection direction and cuts of the unit sphere

Orthogonal projections of multidimensional ellipsoids – II – the surface of the projection image is a lower dimensional ellipsoid

In this post series we presently look at orthogonal projections of an ellipsoid in a n-dimensional (Euclidean) space onto a (n-1)-dimensional subspace. The ellipsoid is a closed surface in the ℝn and has a dimensionality of (n-1). Orthogonal projection means that the target subspace is orthogonal to a line of projection (defined by a projection vector) which is the same… Read More »Orthogonal projections of multidimensional ellipsoids – II – the surface of the projection image is a lower dimensional ellipsoid