Fundamentals of Matrix Calculus

In this introductory chapter, the essential fundamentals of classical matrix calculus are compiled, as outlined in [Zurmuehl/Falk 1986] and [Pestel/Leckie 1963]. The primary objective is to acquaint the reader with index notation for multidimensional matrices, which will serve as the basis for all arithmetic operations in the subsequent context. This introduction aims to make the reader comfortable with this notation, which may seem unfamiliar compared to traditional symbolic notation found in related publications. Another important aspect is the generalization of the conventional notation, primarily limited to one- and two-dimensional matrices, which was originally designed for manual computations on traditional media. Therefore adequate formulations for multidimensional matrices are crucial, especially by use of programming languages in the mathematical-scientific field that handle arrays of several dimensions using standard implementations. This introduction to index formulations also prepares the reader for the subsequent acquaintance with covariant and contravariant tensors in the following chapters on tensor analysis and computational tensor/matrix methods.