Macierze wielowskaźnikowe w naukach ekonomiczno-społecznych

Abstract

The present application of spatial econometrics is still insufficient to launch complex exami- nations of processes in social and economic development. Econometrics employs numerous achievements of mathematical statistics, particularly those connected with multi-variable methods. However, what makes an obstacle in such an operation is two-dimensionality of algebraic objects that both sciences use for their calculations. Therefore, I would like to promote in my work an instrument of multi-index matrices that may significantly accelerate the progress taking place in the research of complex social and economic processes. This work defined the multi-index matrices together with operations that may be carried out on them. Special consideration is given to their products, the definition of which includes not only 5 known and practically applied ones but also 8 others, the application of which is still to be discovered. Using the notion of the associa- tive matrix the following have been defined: (lsc)-non-singular matrix, (lsc)-canonical matrix, (lsc)-equivalence, p-index standard matrix, (lsc)-trace and other notions indispensable for the matrix algebra. The work includes definition of generalized invertible multi-index matrix and theorems connected with that notion, including the existence criteria for H- and L-generalized invertible Alsc matrices as well as the Moore–Penrose inverse. The work includes also other appli- cations of multi-index matrices differing from the statistical methods.