An Efficient Method for Computing the Inverse and Eigenvalues of Circulant Matrices with Lucas Numbers

Abstract

In this article, the inverse including the determinant, and the eigenvalues of circulant matrices with entry Lucas numbers are formulated explicitly in a simple way so that their computations can be constructed efficiently. The formulation method of the determinant and inverse is simply applying the theory of elementary row or column operations and can be unified in one theorem. Meanwhile, for the eigenvalues formulation, the recently known formulation in the case of general circulant matrices is simplified by observing the specialty of the Lucas sequence and applying cyclic group properties of unit circles in the complex plane. Then, an algorithm of those formulations is constructed efficiently. From some  implementation facts also showed that the algorithms performed very fast and was able to calculate large size of circulant matrices.