Introduction to Eigenanalysis

Eigenanalysis and its numerous offsprings form the suite of algebraic operations most important and relevant to data analysis, as well as to dynamical systems, modeling, numerical analysis, and related key branches of applied mathematics. This chapter introduces and places in a broader context the algebraic operation of eigen-decomposition. To have eigen-decomposition, a matrix must be square. Yet data matrices are very rarely square. The direct relevance of eigen-decomposition to data analysis is therefore limited. Indirectly, however, generalized eigenanalysis is enormously important to studying data matrices. Because of the centrality of generalized eigenanalysis to data matrices, and because those generalizations build, algebraically and logically, on eigenanalysis itself, it makes sense to discuss eigenanalysis at some length.