Low complexity matrix projections preserving actions on vectors

Cipolla, Stefano, Di Fiore, Carmine and Zellini, Paolo (2019) Low complexity matrix projections preserving actions on vectors. Calcolo, 56 (2), [8]. (doi:10.1007/s10092-019-0305-8).

Record type: Article

Abstract

In this paper we prove that, given a n× 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 L_A the matrix closest to A from L in the Frobenius norm, one has LAjV=AjV 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 that L, for mr≪ n, is a low complexity matrix algebra. The new theoretical results here introduced allow to investigate new possible preconditioners L_A for the Conjugate Gradient method and new quasi-Newton algorithms suitable to solve large scale optimization problems.

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Accepted/In Press date: 9 March 2019

e-pub ahead of print date: 19 March 2019

Additional Information: Funding Information: C.D.F. was partially supported by MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

Keywords: Arnoldi method, Block-Krylov spaces, Direction preserving projections, Matrix projections, Unitary decomposition by householder matrices