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Abstract and Applied Analysis
Volume 2015, Article ID 329329, 11 pages
http://dx.doi.org/10.1155/2015/329329
Research Article

VanderLaan Circulant Type Matrices

Department of Mathematics, Linyi University, Linyi, Shandong 276000, China

Received 15 August 2014; Accepted 14 October 2014

Academic Editor: Shen Yin

Copyright © 2015 Hongyan Pan and Zhaolin Jiang. 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.

Abstract

Circulant matrices have become a satisfactory tools in control methods for modern complex systems. In the paper, VanderLaan circulant type matrices are presented, which include VanderLaan circulant, left circulant, and -circulant matrices. The nonsingularity of these special matrices is discussed by the surprising properties of VanderLaan numbers. The exact determinants of VanderLaan circulant type matrices are given by structuring transformation matrices, determinants of well-known tridiagonal matrices, and tridiagonal-like matrices. The explicit inverse matrices of these special matrices are obtained by structuring transformation matrices, inverses of known tridiagonal matrices, and quasi-tridiagonal matrices. Three kinds of norms and lower bound for the spread of VanderLaan circulant and left circulant matrix are given separately. And we gain the spectral norm of VanderLaan -circulant matrix.