Low complexity matrix projections preserving actions on vectors

In this paper we prove that, given a n x n symmetric matrix A, a matrix V with r orthonormal columns and an integer m = 1, mr = n, it is possible to devise a matrix algebra L such that, denoting by LA the matrix closest to A from L in the Frobenius norm, one has Lj AV = Aj V for j = 0,..., m -1. The algebra L is the space of all matrices that are diagonalized by a given orthogonal matrix L. We show, moreover, that L can be constructed as the product of mr Householder matrices, so thatL, formr n, is a low complexity matrix algebra. The new theoretical results here introduced allow to investigate newpossible preconditioners LA for the ConjugateGradient method and new quasi-Newton algorithms suitable to solve large scale optimization problems.