Skip to content
ellipse and circles through end points of the semi-axes

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 ….

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 ….

projections of ellipsoid from 4 to 3 dimensions

Orthogonal projections of multidimensional ellipsoids – VII – experimental checks of projections from a 4-dimensional space to sub-spaces

This post series covers the topic of orthogonal projections of ellipsoids from their embedding multidimensional space down to sub-spaces. In this post we will verify some of our mathematical results by a numerical experiment: We will compute the projections of an ellipsoid from its embedding 4-dimensional (Euclidean) space down to 3- and 2-dimensional sub-spaces. We will check with the help… Read More »Orthogonal projections of multidimensional ellipsoids – VII – experimental checks of projections from a 4-dimensional space to sub-spaces

Projection of ellipsoid form 4 to 2 dimensions

Orthogonal projections of multidimensional ellipsoids – VI – considerations and vector generation for numerical experiments in four dimensions

After our recent excursion to Schur complements, some readers may wish to get a visual confirmation of our rather theoretical results. As usual in this blog, we will perform some numerical experiments. To get started, I will restrict my efforts to the projection of an ellipsoid from a 4-dimensional Euclidean space down to three- and two-dimensional sub-spaces. In a later… Read More »Orthogonal projections of multidimensional ellipsoids – VI – considerations and vector generation for numerical experiments in four dimensions

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