Symmetrical Multidimensional Matrices and Their Inversion

The article discusses the development of the theory of the multidimensional matrices in the part regarding the symmetrical multidimensional matrices. The symmetry property is considered in terms of the structure of the multidimensional matrix. The so called kq-dimensional symmetrical matrices are considered along with the multidimensional symmetrical matrices, i. e. the matrices symmetrical to the multi-indices containing q indices. Unit, unit symmetrical and unit kq-symmetric matrices are considered. The matrices inverse to the multidimensional matrices with respect to the unit, unit symmetrical, and unit kq-symmetrical matrices are defined. It is proven that the matrices, inverse to the multidimensional matrices with respect to the unit symmetrical and unit kq-symmetrical matrices are the Moore–Penrose matrices. The distinct instances are given.