A Note on the Minimum Cardinality of Critical Sets of Inertias for Irreducible Zero-nonzero Patterns of Order 4
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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.
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References
- R. A. Horn, C. R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1995.
- 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.
- 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.
- M. S. Cavers, K. N. Vander Meulen, Inertially arbitrary nonzero patterns of order 4, Electron. J. Linear Algebra 16(2007) 30-43.