Intuitive derivation of loop inverses and array algebra |
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| Authors: | Urho A Rauhala |
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| Institution: | (1) Geodetic Services, Inc., P.O. Box 3668, 32903 Indialantic, Florida |
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| 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
by a parameter exchangeX→L
0, where
X is a set of unknown observables,A
0 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
expressed in observed values
. The loop inverses are found by the back substitution expressing ∧X in terms ofL through
. 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 A
0
−1
(AA
0
−1
)ℓ reduces into the ℓ-inverse Aℓ=(ATA)−1AT. The physical interpretation of the design matrixA A
0
−1
as an interpolator, associated with the parametersL
0, and the consideration of its multidimensional version has resulted in extended rules of matrix and tensor calculus and mathematical
statistics called array algebra. |
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| Keywords: | |
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