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ON NILPOTENCE INDICES OF SIGN PATTERNS

  • Published : 2010.01.31

Abstract

The work in this paper was motivated by [3], where Eschenbach and Li listed four 4 by 4 sign patterns, conjectured to be nilpotent sign patterns of nilpotence index at least 3. These sign patterns with no zero entries, called full sign patterns, are shown to be potentially nilpotent of nilpotence index 3. We also generalize these sign patterns of order 4 so that we provide classes of n by n sign patterns of nilpotence indices at least 3, if they are potentially nilpotent. Furthermore it is shown that if a full sign pattern A of order n has nilpotence index k with $2{\leq}k{\leq}n-1$, then sign pattern A has nilpotent realizations of nilpotence indices k, k + 1, $\ldots$, n. Hence, the four 4 by 4 sign patterns in [3, page 91] also allow nilpotent realizations of nilpotence index 4.

Keywords

Jordan block;nilpotence index;potentially nilpotent sign pattern

References

  1. R. A. Brualdi and B. L. Shader, Matrices of Sign-solvable Linear Systems, Cambridge University Press, New York, 1995.
  2. M. S. Cavers and K. V. Vander Meulen, Spectrally and inertially arbitrary sign patterns, Linear Algebra and its Applications 394 (2005), 53–72. https://doi.org/10.1016/j.laa.2004.06.003
  3. C. A. Eschenbach and Z. Li, Potentially nilpotent sign pattern matrices, Linear Algebra and its Applications 299 (1999), 81–99. https://doi.org/10.1016/S0024-3795(99)00163-9
  4. Y. Gao, Z. Li, and Y. Shao, Sign patterns allowing nilpotence of index 3, Linear Algebra and its Applications 424 (2007), 55–70. https://doi.org/10.1016/j.laa.2006.03.038
  5. I.-J. Kim, D. D. Olesky, B. L. Shader, P. van den Driessche, H. Van Der Holst, and K. N. Vander Meulen, Generating potentially nilpotent full sign patterns, Electronic Journal of Linear Algebra 18 (2009), 162–175.

Cited by

  1. Tree sign patterns that allow nilpotence of index 4 vol.63, pp.5, 2015, https://doi.org/10.1080/03081087.2014.914930
  2. Potentially nilpotent patterns and the Nilpotent-Jacobian method vol.436, pp.12, 2012, https://doi.org/10.1016/j.laa.2011.05.017