The inverse matrix of some circulant matrices

TL;DR

This paper provides necessary and sufficient conditions for the invertibility of certain circulant and symmetric matrices, explicitly computes their inverses, and reduces computational complexity using boundary value problem techniques.

Contribution

It introduces new criteria for invertibility and explicit inverse formulas for a class of circulant and symmetric matrices, improving computational efficiency.

Findings

Derived invertibility conditions for parameter-dependent circulant matrices

Explicit formulas for inverses of matrices with arithmetic or geometric sequences

Characterized invertibility of symmetric tridiagonal matrices

Abstract

We present here necessary and sufficient conditions for the invertibility of circulant and symmetric matrices that depend on three parameters and moreover, we explicitly compute the inverse. The techniques we use are related with the solution of boundary value problems associated to second order linear difffference equations. Consequently, we reduce the computational cost of the problem. In particular, we recover the inverses of some well known circulant matrices whose coeffifficients are arithmetic or geometric sequences, Horadam numbers among others. We also characterize when a general symmetric circulant and tridiagonal matrix is invertible and in this case, we compute explicitly its inverse.