Dietrich Trenkler

On the Product of Rotations.

Using the elementary tools of matrix theory, we show that the product of two rotations in the three-dimensional Euclidean space is a rotation again. For this purpose, three types of rotation matrices are identified which are of simple structure. One of them is the identity matrix, and each of the other two types can be uniquely characterized by exactly one vector. The resulting products are investigated by using the basic properties of the vector cross product.

A simple parameterization of 3×3 magic squares

In this article a new parameterization of magic squares of order three is presented. This parameterization permits an easy computation of their inverses, eigenvalues, eigenvectors and adjoints. Some attention is paid to the Luoshu, one of the oldest magic squares.

On the euclidean distance between biased estimators

The purpose of this paper is to extend the results achieved by Hsuan (1981) to a wide class of biased estimators. It is shown that in the case of multicollinearity ridge and Iteration estimator can be made very close to the principal component estimator whereas the shrunken estimator does not have this property.

Intersection of three planes revisited – an algebraic approach

ABSTRACT Given three planes in space, a complete characterization of their intersection is provided. Special attention is paid to the case when the intersection set does not exist of one point only. Besides the vector cross product, the tool of generalized inverse of a matrix is used extensively.

Polar and singular value decomposition of 3×3 magic squares

In this note, we find polar as well as singular value decompositions of a 3×3 magic square, i.e. a 3×3 matrix M with real elements where each row, column and diagonal adds up to the magic sum s of the magic square.

The Numerical Range of the Luoshu Is a Piece of Cake--Almost.

Summary The numerical range, easy to understand but often tedious to compute, provides useful information about a matrix. Here we describe the numerical range of a 3 × 3 magic square. Applying our results to one of the most famous of those squares, the Luoshu, it turns out that its numerical range is a piece of cake—almost.