In this work, an alternative, data-driven fault diagnosis (FDI)-framework based on symmetric, positive-definite (SPD) matrices will be introduced. In fault diagnosis and control theory, SPD-matrices serve important information about the system under investigation depending on the analysis and interpretation. Taking into account their special characteristics and for the purpose of an alternative way of investigation, the basic concepts of Riemannian Geometry are introduced as a mathematical tool. The fundamental SPD-matrix-based FD-scheme is characterised by a flexible implementation and independence of statistical distribution of data. The article also delivers an overview about the possible realisations in model-based and data-driven FDI as well as in the area of machine learning (ML). Further, a new modelling of a linear, time-invariant system will be introduced and extended to an FD scheme.