Received: 29 May 2017 Revised: 26 October 2018 Accepted: 15 November 2018 DOI: 10.1002/nla.2225 RESEARCH ARTICLE Regularizing properties of a class of matrices including the optimal and the superoptimal preconditioners Stefano Cipolla 1 Carmine Di Fiore 2 Fabio Durastante 3 Paolo Zellini 2 1 Department of Mathematics, University of Padova, Padua, Italy 2 Department of Mathematics, University of Rome Tor Vergata, Rome, Italy 3 Department of Computer Science, University of Pisa, Pisa, Italy Correspondence Stefano Cipolla, Department of Mathematics, University of Padova, via Trieste 33, 35121 Padua, Italy. Email: cipolla@math.unipd.it Funding information Gruppo Nazionale per il Calcolo Scientifico dell'Istituto Nazionale di Alta Matematica (INdAM-GNCS) ; 2018 INdAM-GNCS project “Tecniche innovative per problemi di algebra lineare”; MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, Grant/Award Number: CUP E83C18000100006 Summary In this work, given a positive definite matrix A, we introduce a class of matrices related to A, which is obtained by suitably combining projections of its powers onto algebras of matrices simultaneously diagonalized by a unitary transform. After a detailed theoretical study of some spectral properties of the matrices of this class, which suggests their use as regularizing preconditioners, we prove that such matrices can be cheaply computed when the matrix A has a Toeplitz structure. We provide numerical evidence of the advantages coming from the employment of the proposed preconditioners when used in regularizing procedures. KEYWORDS optimal preconditioning, regularizing preconditioners, superoptimal preconditioning 1 INTRODUCTION Given a matrix A ∈ ℂ n×n and  ∶= sd U ={Ud(z)U ∗ ∶ z ∈ ℂ n }, with d(z) being the diagonal matrix with entries z i and U being a unitary fixed matrix, let us define the optimal preconditioners of A as  A ∶= arg min X∈ ||X - A|| F . Opti- mal circulant preconditioners (i.e., U is the Fourier matrix) have been introduced in the work of T. Chan 1 and studied in the works of R. Chan 2 and Tyrtyshnikov. 3 Superoptimal circulant preconditioners of A—obtained solving the problem min X∈ ||AX - I || F —have been introduced in the work of Tyrtyshnikov 3 and studied in the works of Chan et al. 4 and Di Benedetto et al. 5 There is a huge amount of literature studying the properties of these preconditioners and both have been used in the context of several applications; see, for example, the works of Chan et al. 6 and Bertaccini et al., 7 and the numerous references therein. In the work of Di Benedetto et al., 5 it has been proved that the circulant superopti- mal preconditioners for A = Toeplitz matrix provide a poor approximation of the original Toeplitz matrix when this is ill conditioned: If the considered Toeplitz matrix has a family of eigenvalues collapsing to zero, then the corresponding eigenvalues of the superoptimal preconditioner are well separated from zero; this peculiarity of the superoptimal pre- conditioners causes a slowdown of the preconditioned conjugate gradient (PCG) method when solving exactly the linear system. In contrast, the inability of the superoptimal preconditioner to accurately approximate the eigenvalues collaps- ing to zero is a particularly appreciated peculiarity in the field of image restoration and inverse problems where problems Numer Linear Algebra Appl. 2018;e2225. wileyonlinelibrary.com/journal/nla © 2018 John Wiley & Sons, Ltd. 1 of 17 https://doi.org/10.1002/nla.2225