Skip to content

quadratic form

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

How to compute confidence ellipses – II – equivalence of Schelp’s basic construction method for confidence ellipse with other approaches

In the 1st post of this series, I have motivated a simple method for constructing confidence ellipses for assumedly Bivariate Normal Distributions [BVD] or at least approximate BVDs. A reader has asked me, whether one can prove more rigidly that the proposed method of C. Schelp is equivalent to other BVD-based methods. Well, in this post we show that the… Read More »How to compute confidence ellipses – II – equivalence of Schelp’s basic construction method for confidence ellipse with other approaches

Cholesky decomposition of an ellipse-defining symmetric matrix

An ellipse can be defined via a symmetric, invertible and positive-definite (2×2)-matrix Aq. Such a matrix provides a quadratic form which in turn correlates the components of position vectors to points on an ellipse. This post shows that a Cholesky decomposition of the inverse of Aq provides a method to create an ellipse from a simple set of vectors which… Read More »Cholesky decomposition of an ellipse-defining symmetric matrix

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