Minimal Critical Sets of Refined Inertias for Irreducible Sign Patterns of Order 2

Author(s)
Ber-Lin Yu

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 *S*is 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.

B. Yu, "Minimal Critical Sets of Refined Inertias for Irreducible Sign Patterns of Order 2,"

References

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[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

[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