The last months I have seldom written posts in this blog. The reason is that I am occupied with a book about “The geometry of Multivariate Normal Distributions”. Which will cover a lot more topics than the ones discussed in this blog so far. One of those topics is the use of quadratic form functions as graph functions to produce… Read More »Quadratic form functions as graph functions – I – level sets and gradients
Some of my readers know that I am presently writing a book about Multivariate Normal distributions and related geometric properties of their level sets (multidimensional ellipsoids). For certain topics of the book, I sometimes use the latest free edition of ChatGPT to verify some claims and to get a list of respective papers. Sometimes also for a proof … However,… Read More »Getting the semi-axes of an ellipse by differentiation of its graph functions – a simple step for a human, but not for ChatGPT 5.x ….
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
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