Construction Methods for Sign Patterns Allowing Nilpotence of Index k

Authors: Jun Luo

Abstract:

In this paper, the smallest such integer k is called by the index (of nilpotence) of B such that Bk = 0. In this paper, we study sign patterns allowing nilpotence of index k and obtain four methods to construct sign patterns allowing nilpotence of index at most k, which generalizes some recent results.

Keywords: Sign pattern, Nilpotence, Jordan block.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1336020

References:

[1] Yubin Gao, Zhongshan Li and Yanling Shao, Sign patterns allowing nilpotence of index 3, Linear Algebra Appl., 424 (2007): 55-70.
[2] R. A. Brualdi and B. L. Shader, Matrices of Sign-Solvable Linear Systems, Cambridge University Press, 1995
[3] Marina Arav, Frank J. Hall, Selcuk Koyuncu, Zhongshan Li and Bhaskara Rao, Rational realizations of the minimum rank of a sign pattern matrix, Linear Algebra Appl., 409 (2005): 111-125.
[4] C. A. Eschenbach and Z. Li, Potentially nilpotent sign pattern matrices, Linear Algebra Appl., 299 (1999): 81-99.
[5] L. Yeh, Sign pattern matrices that allow a nilpotent matrix, Bull Aust. Math. Soc., 53 (1996): 189-196.
[6] G. MacGillivray, R. M. Tifenbach and P. van den Driessche, Spectrally arbitrary star sign patterns, Linear Algebra Appl., 400 (2005): 99-119.
[7] M. Catral, D.D. Olesky, P. van den Driessche, Allow problems concerning spectral properties of sign pattern matrices: A survey, Linear Algebra Appl., 430 (2009): 3080-3094.
[8] R. A. Brualdi and H. J. Ryser, Combinatorial Matrix Theory, Cambridge University Press, Cambridge, 1991.