**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**32722

##### Minimal Critical Sets of Inertias for Irreducible Zero-nonzero Patterns of Order 3

**Authors:**
Ber-Lin Yu,
Ting-Zhu Huang

**Abstract:**

If there exists a nonempty, proper subset S of the set of all (n + 1)(n + 2)/2 inertias such that S Ôèå i(A) is sufficient for any n × n zero-nonzero pattern A to be inertially arbitrary, then S is called a critical set of inertias for zero-nonzero patterns of order n. If no proper subset of S is a critical set, then S is called a minimal critical set of inertias. In [3], Kim, Olesky and Driessche identified all minimal critical sets of inertias for 2 × 2 zero-nonzero patterns. Identifying all minimal critical sets of inertias for n × n zero-nonzero patterns with n ≥ 3 is posed as an open question in [3]. In this paper, all minimal critical sets of inertias for 3 × 3 zero-nonzero patterns are identified. It is shown that the sets {(0, 0, 3), (3, 0, 0)}, {(0, 0, 3), (0, 3, 0)}, {(0, 0, 3), (0, 1, 2)}, {(0, 0, 3), (1, 0, 2)}, {(0, 0, 3), (2, 0, 1)} and {(0, 0, 3), (0, 2, 1)} are the only minimal critical sets of inertias for 3 × 3 irreducible zerononzero patterns.

**Keywords:**
Permutation digraph,
zero-nonzero pattern,
irreducible pattern,
critical set of inertias,
inertially arbitrary.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1077569

**References:**

[1] F. Hall, Z. Li, Sign pattern matrices, in: L. Hogben(Ed.), Handbook of Linear Algebra, Chapman & Hall/CRC Press, Boca Ration, 2007.

[2] R. A. Horn, C. R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1995.

[3] I. J. Kim, D. D. Olesky, P. van den Driessche, Critical sets of inertias for matrix patterns, Linear Multilinear Al. 57(3)(2009) 293-306.

[4] L. M. DeAlba, I. R. Hentzel, L. Hogben, J. McDonald, R. Mikkelson, O. Pryporova, B. Shader, K. N. Vander Meulen, Spectrally arbitrary patterns: reduciblity and the 2n conjecture for n = 5, Linear Algebra Appl. 423(2007) 262-276.

[5] L. Corpuz, J. J. McDonald, Spectrally arbitrary zero-nonzero patterns of order 4, Linear Multilinear Al. 55(3)(2007) 249-273.

[6] M. S. Cavers, K. N. Vander Meulen, Inertially arbitrary nonzero patterns of order 4, Electron. J. Linear Al. 16(2007) 30-43.

[7] I. J. Kim, J. J. McDonald, D. D. Olesky, P. van den Driessche, Inertias of zero-nonzero patterns, Linear Multilinear Al. 55(3)(2007) 229-238.