Grundlagen der Matrizenrechnung

In this introductory chapter the most important basics of the classical matrix calculus are compiled, as found in [Zurmühl/Falk 1986], [Pestel/Leckie 1963]. The primary goal is to introduce the reader to the index notation for multidimensional matrices in order to base all arithmetic operations on it in the following context. The reader may become sufficiently familiar with this notation and gets used to the possibly strange notation, which differs considerably from the traditional symbolic notation of related publications. The second important aspect is the generalization of the conventional notation restricted mostly to one- and two-dimensional matrices, which was originally introduced for computational schemes on traditional writing media. Because adequate formulations for multidimensional matrices are the important basis for all further considerations. This is also due to the use of programming languages in the mathematical-scientific field which are processing arrays of several dimensions by standard implementations. The introduction to index formulations prepares the reader as well for the corresponding acquaintances with covariant and contravariant tensors in the following chapters about tensor analysis and computational tensor/matrix methods.