Some useful features of ellipses
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 ….
During the last posts, I have discussed properties of ellipses and ways to (re-) construct them from elements of a symmetric, invertible and positive-definite (2×2)-matrix, which defines a quadratic form. In the context of Machine Learning we often have to determine confidence ellipses from elements of a numerically determined variance-covariance matrix of statistical bivariate vector-distributions. Formulas relating the geometric properties… Read More »Ellipses via matrix elements – II – numerical tests of formulas
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