Improving approximate inverses based on Frobenius norm minimization

Approximate inverses, based on Frobenius norm minimization, of real nonsingular matrices are analyzed from a purely theoretical point of view. In this context, this paper provides several sufficient conditions, that assure us the possibility of improving (in the sense of the Frobenius norm) some given approximate inverses. Moreover, the optimal approximate inverses of matrix A∈Rn×nA∈Rn×n, among all matrices belonging to certain subspaces of Rn×nRn×n, are obtained. Particularly, a natural generalization of the classical normal equations of the system Ax=bAx=b is given, when searching for approximate inverses N≠AT such that AN   is symmetric and ‖AN-I‖F<‖AAT-I‖FAN-IF