Summary: Spectral Propertiers of Sign Patterns
A n × n sign pattern S is inertially arbitrary if any ordered triple of non-negative integers
( i+ , i-, i0 ) with sum n is the inertia of a matrix with sign pattern S. Sign pattern S is
spectrally arbitrary if for any real monic polynomial p(x) of degree n there is a matrix with sign
pattern S and characteristic polynomial p(x).
Many classes of sign patterns are known to be spectrally or inertially arbitrary but there are no
known necessary and sufficient conditions for this property. The implicit function theorem provides
a sufficient condition and is used to produce many of the known classes. Many fundamental
questions in sign pattern theory are still to be resolved; for example, is it possible to have a
spectrally arbitrary pattern with non-spectrally arbitrary super-patterns?
Patterns whose graph is a tree have more `tractable' qualities which are used to characterize
spectral properties of star sign patterns. A specific pattern, Tn, whose graph is a path, is known
to be spectrally arbitrary for n 7, but the general case is yet be solved.