| dc.description.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. | pl_PL |
