Back
 ALAMT  Vol.3 No.2 , June 2013
Minimal Critical Sets of Refined Inertias for Irreducible Sign Patterns of Order 2
Abstract: Let S be a nonempty, proper subset of all refined inertias. Then, S is called a critical set of refined inertias for ireducible sign patterns of order n if is sufficient for any sign pattern A to be refined inertially arbitrary. If no proper subset of Sis a critical set of refined inertias, then S is a minimal critical set of refined inertias for sign patterns of order n . In this paper, all minimal critical sets of refined inertias for irreducible sign patterns of order 2 are identified. As a by-product, a new approach is presented to identify all minimal critical sets of inertias for irreducible sign patterns of order 2.
Cite this paper: B. Yu, "Minimal Critical Sets of Refined Inertias for Irreducible Sign Patterns of Order 2," Advances in Linear Algebra & Matrix Theory, Vol. 3 No. 2, 2013, pp. 7-10. doi: 10.4236/alamt.2013.32002.
References

[1]   F. Hall and Z. Li, “Sign Pattern Matrices,” In: L. Hogben, Ed., Handbook of Linear Algebra, Chapman & Hall/CRC Press, Boca Ration, 2007.

[2]   R. A. Horn and C. R. Johnson, “Matrix Analysis,” Cambridge University Press, New York, 1985. doi:10.1017/CBO9780511810817

[3]   L. Deaett, D. D. Olesky and P. van den Driessche, “Refined Inertially and Spectrally Arbitrary Zero-Nonzero Patterns,” The Electronic Journal of Linear Algebra, Vol. 20, 2010, pp. 449-467.

[4]   M. S. Cavers and K. N. Vander Meulen. “Spectrally and Inertially Arbitrary Sign Patterns,” Linear Algebra and Its Applications, Vol. 394, No. 1, 2005, pp. 53-72. doi:10.1016/j.laa.2004.06.003

[5]   M. S. Cavers, K. N. Meulen and L. Vanderspek. “Sparse Inertially Arbitrary Patterns,” Linear Algebra and Its Applications, Vol. 431, No. 11, 2009, pp. 2024-2034. doi:10.1016/j.laa.2009.06.040

[6]   I. J. Kim, D. D. Olesky and P. van den Driessche, “Critical Sets of Inertias for Matrix Patterns,” Linear and Multilinear Algebra, Vol. 57, No. 3, 2009, pp. 293-306. doi:10.1080/03081080701616672

[7]   B. L. Yu, T. Z. Huang, J. Luo and H. B. Hua, “Critical Sets of Refined Inertias for Irreducible Zero-Nonzero Patterns of Orders 2 and 3,” Linear Algebra and Its Applications, Vol. 437, No. 2, 2012, pp. 490-498. doi:10.1016/j.laa.2012.03.007

 
 
Top