Abstract
Abstract Explicit expressions for the coefficients of the group inverse of a circulant matrix depending on four complex parameters are analytically derived. The computation of the entries of the group inverse are now reduced to the evaluation of a polynomial. Moreover, our methodology applies to both the invertible and the singular case, the latter being computationally less expensive. The techniques we use are related to the solution of boundary value problems associated with second order linear difference equations.
Highlights
Introduction and PreliminariesThe problem of solving a linear system with circulant coe cient matrices appears in many problems related to the periodicity of that problem
In [2], we computed the inverse matrix of some circulant matrices of order n ≥ with three real parameters at most and in [3] we considered complex parameters and we determined the expression for their group inverse
We reduced signi cantly the computational cost of applying Lemma 2.1, since the key point for nding the mentioned inverse matrix consists in solving the system that provides g(a)τ by means of a non-homogeneous rst order linear di erence equation
Summary
The problem of solving a linear system with circulant coe cient matrices appears in many problems related to the periodicity of that problem. This kind of system occurs in many applications: time series analysis, image processing, spline approximation, di erence solutions of partial di erential equations or the nite di erence method to approximate elliptic equations with periodic boundary conditions, see for instance [4]. R. Searle in [12], provided a method for obtaining analytic expressions for the coe cients of the inverse matrix of a family of three-element circulant matrices; O. C, d) and we obtain analytical expressions for the coe cients of their inverse or group inverse. This means that, by just checking some relations between the four coe cients, we can explicitly com-
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