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Ellipsoids – Quadratic Forms

Posts on the math of ellipsoids defined by quadratic forms

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

In this post series we 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 vector) which is the same for all… Read More »Orthogonal projections of multidimensional ellipsoids – II – the surface of the projection is a lower dimensional ellipsoid

orthogonal projections of ellipsoid

Orthogonal projections of multidimensional ellipsoids – I – points on the ellipsoid that give us the surface points of the projection

A reader of a parallel post-series on n-ellipsoids has asked me how I could know what the matrix for the quadratic form of the orthogonal projection of a (n-1) dimensional-ellipsoid onto a (n-1)-dimensional sub-space of the ℝn looks like. I had stated in previous posts that we can directly derive the elements of the matrix describing projections of ellipsoids or… Read More »Orthogonal projections of multidimensional ellipsoids – I – points on the ellipsoid that give us the surface points of the projection

Surfaces of n-dimensional ellipsoids – I – quadratic form and matrix equation

Multidimensional ellipsoids are mathematically interesting figures per se. But there is a reason why they sometimes also appear in the context of Machine Learning experiments. One reason is that Multivariate Normal Distributions [MND] often describe the statistical distributions of properties which characterize natural objects we investigate by ML-methods. And the locations of constant probability density of MNDs are surfaces of… Read More »Surfaces of n-dimensional ellipsoids – I – quadratic form and matrix equation