**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**31819

##### 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:**

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

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[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.