Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 30054
A Note on the Minimum Cardinality of Critical Sets of Inertias for Irreducible Zero-nonzero Patterns of Order 4

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 [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

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF

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.