Intuitive derivation of loop inverses and array algebra

Abstract

Array algebra forms the general base of fast transforms and multilinear algebra making rigorous solutions of a large number (millions) of parameters computationally feasible. Loop inverses are operators solving the problem of general matrix inverses. Their derivation starts from the inconsistent linear equations\(\mathop A\limits_{m n} \mathop X\limits_{n ,1} \ne \mathop L\limits_{m ,1} \) by a parameter exchangeX→L0, where\(\mathop {L_0 }\limits_{p , 1} = \mathop {A_0 }\limits_{p n} \mathop X\limits_{n , 1} \)X is a set of unknown observables,A0 forming a basis of the so called “problem space”. The resulting full rank design matrix of parameters L0 and its l-inverse reveal properties speeding the computational least squares solution\(\mathop {\hat L_0 }\limits_{p , 1} \) expressed in observed values\(\mathop L\limits_{m , 1} \). The loop inverses are found by the back substitution expressing ∧X in terms ofL through\(\hat L_0 \). Ifp=rank (A) ≤n, this chain operator creates the pseudoinverseA+. The idea of loop inverses and array algebra started in the late60's from the further specialized case,p=n=rank (A), where the loop inverse A0−1(AA0−1)l reduces into the l-inverse Al=(ATA)−1AT. The physical interpretation of the design matrixA A0−1 as an interpolator, associated with the parametersL0, and the consideration of its multidimensional version has resulted in extended rules of matrix and tensor calculus and mathematical statistics called array algebra.