The construction of optimal preconditioners for linear systems is discussed. For the construction of the preconditioner of the preconditioned system , , the minimization problem is considered, where denotes the Frobenius norm and is a subspace of the space of all matrices with real coefficients .
In a more general framework, the author analyses the problem with an arbitrary subspace of the space . At first some spectral properties of the solution are established. Then these results are applied to analyse the effectiveness of the approximate inverse preconditioner . The main result is the following: When the smallest singular value or the smallest eigenvalue’s modulus of the matrix increases to 1 the effectiveness of the preconditioner improves.