Representation of Associated Data by Matrix Operators

Abstract

It is shown that associated pairs of vectoral items (Q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(r)</sup> , X <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(r)</sup> ) can be recorded by transforming them into a matrix operator M so that a particular stored vector X <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(r)</sup> can be reproduced by multiplying an associated cue vector Q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(r)</sup> by M. If the number of pairs does not exceed the dimension of the cue and all cue vectors are linearly independent, then the recollections are perfect replicas of the recorded items and there will be no crosstalk from the other recorded items. If these conditions are not valid, the recollections are still linear least square approximations of the X <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(r)</sup> . The relationship of these mappings to linear estimators is discussed. These transforms can be readily implemented by linear analog systems.