Intuitive derivation of loop inverses and array algebra: Fid-Bau Portal

Intuitive derivation of loop inverses and array algebra

Rauhala, Urho A.

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 $ L_{0} $ 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 $ A^{ℓ} $=($ A^{T} $A)−1$ A^{T} $. 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.