A FAST COMPUTATION FOR EIGENVALUES OF CIRCULANT MATRICES WITH ARITHMETIC SEQUENCE

In this article, we derive simple formulations of the eigenvalues, determinants, and also the inverse of circulant matrices whose entries in the first row form an arithmetic sequence. The formulation of the determinant and inverse is based on elementary row and column operations transforming the matrix to an equivalent diagonal matrix so that the formulation is obtained easily. Meanwhile, for the eigenvalues formulation, we simplify the known result of formulation for the general circulant matrices by exploiting the properties of the cyclic group induced by the set of all roots of  as the set of points in the unit circle in the complex plane, and also by considering the specific property of arithmetic sequence. Then, we construct an algorithm for the eigenvalues formulation. This algorithm shows a better computation compared to the previously known result for the general case of circulant matrices

Networked systems with incomplete information

Copyright © 2015 Zidong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.In this special issue, we have solicited submissions from electrical engineers, control engineers, computer scientists, and mathematicians. After a rigorous peer review process, 18 papers have been selected that provide overviews, solutions, or early promises, to manage, analyse, and interpret dynamical behaviours of networked systems. These papers have covered both the theoretical and practical aspects of networked system with incomplete information in the broad areas of dynamical systems, mathematics, statistics, operational research, and engineering

Exact Determinants of Some Special Circulant Matrices Involving Four Kinds of Famous Numbers

Circulant matrix family is used for modeling many problems arising in solving various differential equations. The RSFPLR circulant matrices and RSLPFL circulant matrices are two special circulant matrices. The techniques used herein are based on the inverse factorization of polynomial. The exact determinants of these matrices involving Perrin, Padovan, Tribonacci, and the generalized Lucas number are given, respectively

On the Explicit Formula for Eigenvalues, Determinant, and Inverse of Circulant Matrices

Determining eigenvalues, determinants, and inverse for a general matrix is computationally hard work, especially when the size of the matrix is large enough. But, if the matrix has a special type of entry, then there is an opportunity to make it much easier by giving its explicit formulation. In this article, we derive explicit formulas for determining eigenvalues, determinants, and inverses of circulant matrices with entries in the first row of those matrices in any formation of a sequence of numbers. The main method of our study is exploiting the circulant property of the matrix and associating it with cyclic group theory to get the results of the formulation. In every discussion of those concepts, we also present some computation remarks

Some formulas for determinants of tridiagonal matrices in terms of finite generalized continued fractions: Formulas for determinants of tridiagonal matrices

In the paper, by virtue of induction and properties of determinants, the authors discover explicit and recurrent formulas of evaluations for determinants of general tridiagonal matrices in terms of finite generalized continued fractions and apply these formulas to evaluations for determinants of the Sylvester matrix and two Sylvester type matrices.In the paper, by virtue of induction and properties of determinants, the authors discover explicit and recurrent formulas of evaluations for determinants of general tridiagonal matrices in terms of finite generalized continued fractions and apply these formulas to evaluations for determinants of the Sylvester matrix and two Sylvester type matrices

Exact Inverse Matrices of Fermat and Mersenne Circulant Matrix

The well known circulant matrices are applied to solve networked systems. In this paper, circulant and left circulant matrices with the Fermat and Mersenne numbers are considered. The nonsingularity of these special matrices is discussed. Meanwhile, the exact determinants and inverse matrices of these special matrices are presented