Inertias of zero–nonzero patterns

Abstract

An n by n zero–nonzero pattern is a matrix with entries ∈{*, 0} where * denotes a nonzero real number. If allows all possible inertias, then is inertially arbitrary. It is shown that there exists a reducible n by n inertially arbitrary zero–nonzero pattern with 2n−1 nonzero entries for each n ≥ 6; and that for n = mt with t ≥ 6 and m ≥ 1, there exists a reducible n by n inertially arbitrary zero–nonzero pattern with 2n−m nonzero entries. These reducible inertially arbitrary zero–nonzero patterns are direct sums of irreducible zero–nonzero patterns, one of which is not inertially arbitrary. Furthermore, for these inertially arbitrary zero–nonzero patterns, it is shown that a superpattern need not be inertially arbitrary, these zero–nonzero patterns do not allow all possible spectra, and there are no inertially arbitrary sign patterns having these zero–nonzero patterns.

Keywords:

• Inertia
• Inertially arbitrary
• Sign pattern
• Zero–nonzero pattern

Keywords:

• AMS Subject Classifications:15A18
• 15A29
• 05C50

Acknowledgement

Research supported in part by NSERC Discovery Grants.