On tensors and multidimensional matrices in multidimensional probabilistic modeling

Abstract

The analysis of the general concepts both of the tensor and multidimensional-matrix approaches in probabilistic modeling is performed and the relationships between the tensor and the multidimensional matrix are clarified. The matrix representation of the second order tensor definition known in the literature is generalized to the arbitrary order tensors. The theorems on the tensor nature of the covariance matrix and the multidimensional matrix of the arbitrary order moments of the random vector are proved. The theorem on the orthogonality of the transformation matrix of the arbitrary order multidimensional-matrix moment of the random vector provided the orthogonality of the random vector transformation is proved