**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30669

##### A Note on the Minimum Cardinality of Critical Sets of Inertias for Irreducible Zero-nonzero Patterns of Order 4

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

**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 [Kim, Olesky and Driessche, Critical sets of inertias for matrix patterns, Linear and Multilinear Algebra, 57 (3) (2009) 293-306], identifying all minimal critical sets of inertias for n×n zero-nonzero patterns with n ≥ 3 and the minimum cardinality of such a set are posed as two open questions by Kim, Olesky and Driessche. In this note, the minimum cardinality of all critical sets of inertias for 4 × 4 irreducible zero-nonzero patterns is identified.

**Keywords:**
Zero-nonzero pattern,
inertia,
critical set of inertias,
inertially arbitrary

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1075284

**References:**

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

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

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

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