Array Algebra Expansion of Matrix and Tensor Calculus: Part 1

Abstract

Array algebra expands the foundations of linear and nonlinear estimation theories, differential and integral calculus, numerical analysis, and fast transform techniques. It originates from an extension of the two-dimensional Kronecker or tensor products and related operators of the traditional vector, matrix, and tensor calculus using the general theory of matrix inverses called inverses. A summary of the foundations of multilinear array algebra and loop inverse estimation is presented in part 1 of this paper. It is then expanded to include the latest developments in nonlinear estimation and applied mathematics using some unified matrix and tensor operators. The new operators are used in part 2 to derive the general theory of direct solution (one hyper iteration) techniques of rank-deficient nonlinear systems as an expansion of the loop inverse estimators and Q-surface tensor solution.