Matrix Algebras and Displacement Decompositions

Abstract

A class $\xi$ of algebras of symmetric n × n matrices, related to Toeplitz-plus-Hankel structures and including the well-known algebra $\mathcal{H}$ diagonalized by the Hartley transform, is investigated. The algebras of $\xi$ are then exploited in a general displacement decomposition of an arbitrary n × n matrix A. Any algebra of $\xi$ is a 1-space, i.e., it is spanned by n matrices having as first rows the vectors of the canonical basis. The notion of 1-space (which generalizes the previous notions of $\mathcal{L}_1$ space [Bevilacqua and Zellini, Linear and Multilinear Algebra, 25 (1989), pp. 1--25] and Hessenberg algebra [Di Fiore and Zellini, Linear Algebra Appl., 229 (1995), pp. 49--99]) finally leads to the identification in $\xi$ of three new (non-Hessenberg) matrix algebras close to $\mathcal{H}$, which are shown to be associated with fast Hartley-type transforms. These algebras are also involved in new efficient centrosymmetric Toeplitz-plus-Hankel inversion formulas.