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quadratic form

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

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