Several Spectrally Non-Arbitrary Ray Patterns of Order 4

Authors: Ling Zhang, Feng Liu

Abstract:

A matrix is called a ray pattern matrix if its entries are either 0 or a ray in complex plane which originates from 0. A ray pattern A of order n is called spectrally arbitrary if the complex matrices in the ray pattern class of A give rise to all possible nth degree complex polynomial. Otherwise, it is said to be spectrally non-arbitrary ray pattern. We call that a spectrally arbitrary ray pattern A of order n is minimally spectrally arbitrary if any nonzero entry of A is replaced, then A is not spectrally arbitrary. In this paper, we find that is not spectrally arbitrary when n equals to 4 for any θ which is greater than or equal to 0 and less than or equal to n. In this article, we give several ray patterns A(θ) of order n that are not spectrally arbitrary for some θ which is greater than or equal to 0 and less than or equal to n. by using the nilpotent-Jacobi method. One example is given in our paper.

Keywords: Spectrally arbitrary, Nilpotent matrix, Ray patterns, sign patterns.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.3607613

References:

[1] J. H. Drew, C. R. Johnson, D. D. Olesky, P. van den Driessche, “Spectrally arbitrary patterns,” Linear Algebra Appl., vol. 308, pp. 121-137, 2000.
[2] M.S. Cavers and S.M. Fallat, “Allow problems concerning spectral properties of patterns,” Electron. J. Linear Algebra., vol. 23, pp. 731-754 , 2012.
[3] M. Catral, D.D. Olesky, and P. van den Driessche, “Allow problems concerning spectral properties of sign pattern matrices: A survey,” Linear Algebra Appl., vol. 430, pp. 3080-3094 , 2009.
[4] M. Cavers, C. Garnett, I.-J. Kim, D.D. Olesky, P. van den Driessche and K. Vander Meulen, “Techniques for identifying inertially arbitrary patterns,” Electron. J. Linear Algebra., vol. 26, pp. 71-89 , 2013.
[5] L. Elsner, D. Hershkowitz, “On the spectra of close-to-Schwarz matrices,” Linear Algebra Appl., vol. 363, pp.81-88, 2003.
[6] In-JaeKim, Bryan L.Shader, Kevin N. Vander Meulen and Matthew West, “Spectrally arbitrary pattern extensions,” Linear Algebra Appl., vol. 517, pp. 120-128, 2017.
[7] J. J. McDonald, D. D. Olesky, M. J. Tsatsomeros, P. van den Driessche, “On the spectra of striped sign patterns,” Linear and Multilinear Algebra, vol. 51, pp. 39-48, 2003.
[8] A. Behn, K.R. Driessel, I.R. Hentzel, K. Vander Velden and J. Wilson, “ Some nilpotent, tridiagonal matrices with a special sign pattern,” Linear Algebra Appl., vol. 36, no. 12, pp. 4446–4450 , 2012.
[9] M. S. Cavers and K. N. Vander Meulen, “Spectrally and inertially arbitrary sign patterns,” Linear Algebra Appl., vol. 394, pp. 53-72 , 2005.
[10] M. S. Cavers, I. J. Kim, B. L. Shader and K. N. Vander Meulen, “On determining minimal spectrally arbitrary patterns,” Electron. J. Linear Algebra., vol. 13, pp. 240-248 , 2005.
[11] G. MacGillivray, R. M. Tifenbach, P. van den Driessche, “Spectrally arbitrary star sign patterns,” Linear Algebra Appl., vol. 400, pp. 99-119, 2005.
[12] T. Britz, J. J. McDonald, D. D. Olesky and P. van den Driessche, “Minimal spectrally arbitrary sign patterns,” SIAM J. Matrix. Anal. Appl., vol. 36, pp. 257–271, 2004.
[13] L. Corpuz and J.J. McDonald, “Spectrally arbitrary zero nonzero patterns of order 4,” Linear and Multilinear Algebra, vol. 55, pp. 249-273, 2007.
[14] L. M. DeAlba, I. R. Hentzel, L. Hogben, J. J. McDonald, R. Mikkelson and O. Pryporova, “Spectrally arbitrary patterns: Reducibility and the 2n conjecture for n = 5,” Linear Algebra Appl., vol. 423, pp. 262-276, 2007.
[15] Yubin Gao and Yanling Shao, “New classes of spectrally arbitrary ray patterns,” Linear Algebra Appl., vol. 434, pp. 2140-2148, 2011. .
[16] Yinzhen Mei, Yubin Gao, Yan Ling Shao and Peng Wang, “A new family of spectrally arbitrary ray patterns,” Czechoslovak Mathematical Journal., vol. 66, pp. 1049–10589, 2016.
[17] Y. Mei, Y. Gao, Y. Shao and P. Wang, “The minimum number of nonzeros in a spectrally arbitrary ray pattern,” Linear Algebra Appl., vol. 453, pp. 99–109, 2014
[18] Ling Zhang, Ting-Zhu Huang, Zhongshan Li and Jing-Yue Zhang, “Several spectrally arbitrary ray patterns,” Linear and Multilinear Algebra, vol.61, pp. 543-564, 2013.
[19] L. Corpuz and J.J. McDonald, “Spectrally arbitrary zero nonzero patterns of order 4,” Linear and Multilinear Algebra, vol. 55, pp. 249-273 , 2007.