On $k$-circulant matrices involving geometric sequence

Abstract

In this paper we consider a $k$-circulant matrix with geometric sequence, where $k$ is a nonzero complex number. The eigenvalues, the determinant, the Euclidean norm and bounds for the spectral norm of such matrix are investigated. The method for obtaining the inverse of a nonsingular $k$-circulant matrix, was presented in [On $k$-circulant matrices (with geometric sequence), Quaest. Math. 2016]. A generalization of that method is given in this paper, and using it, the inverse of a nonsingular $k$-circulant matrix with geometric sequence is obtained. The Moore-Penrose inverse of a singular $k$-circulant matrix with geometric sequence is determined in a different way than the way using in [On $k$-circulant matrices (with geometric sequence), Quaest. Math. 2016].