An Efficient Method for Computing the Inverse and Eigenvalues of Circulant Matrices with Lucas Numbers
Sugi Guritman
Division of Pure Mathematics, Department of Mathematics, Faculty of Mathematics and Natural Sciences, IPB University, IPB Campus Dramaga , Bogor-16680, Indonesia.
Jaharuddin *
Division of Pure Mathematics, Department of Mathematics, Faculty of Mathematics and Natural Sciences, IPB University, IPB Campus Dramaga , Bogor-16680, Indonesia.
Teduh Wulandari
Division of Pure Mathematics, Department of Mathematics, Faculty of Mathematics and Natural Sciences, IPB University, IPB Campus Dramaga , Bogor-16680, Indonesia.
Siswandi
Division of Pure Mathematics, Department of Mathematics, Faculty of Mathematics and Natural Sciences, IPB University, IPB Campus Dramaga , Bogor-16680, Indonesia.
*Author to whom correspondence should be addressed.
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.
Keywords: Circulant matrix, eigenvalues, determinant, inverse, cyclic group, lucas sequence