Spectrally arbitrary pattern extensions

Abstract

A matrix pattern is often either a sign pattern with entries in {0, +, −} or, more simply, a nonzero pattern with entries in {0, * }. A matrix pattern A is spectrally arbitrary if for any choice of a real matrix spectrum, there is a real matrix having the pattern A and the chosen spectrum. We describe a graphical technique, a triangle extension, for constructing spectrally arbitrary patterns out of some known lower order spectrally arbitrary patterns. These methods provide a new way of viewing some known spectrally arbitrary patterns, as well as providing many new families of spectrally arbitrary patterns. We also demonstrate how the technique can be applied to certain inertially arbitrary patterns to obtain larger inertially arbitrary patterns. We then provide an additional extension method for zero-nonzero patterns, patterns with entries in {0, * , ⊛}.2010 Mathematics Subject Classification. 15A18, 15A29, 15B35.

Counterexamples on spectra of sign patterns

“…The study of spectra of matrix patterns deserved a significant amount of attention in recent publications. The conjecture mentioned in the abstract appeared in one of the foundational papers on this topic ( [10]), and many subsequent works proved it in different special cases ( [3,7,12,14,15]). One of the known sufficient conditions for superpatterns to be spectrally arbitrary is the Nilpotent Jacobian condition ( [2,10]), which allowed to solve several intriguing problems on this topic ( [4,11,19]).…”

Bordering for spectrally arbitrary sign patterns

“…Focusing on the coefficients of p X B (z), we can rewrite p X B (z) as (11) for some polynomials S r,ℓ and T r,i,ℓ in the variable entries of X A . Note that the variables in the last row of X A do not appear in S r,ℓ or T r,i,ℓ .…”

“…Remark 2.2. A method in [11] called triangle extension on arc (u, v) (in the digraph associated with A) is equivalent to a special case of applying Theorem 2.1 to row u of A and entry (u, v), namely in the situation that a uu and a uv are the only nonzero entries in row u of A.…”

“…Note that since row 4 of A has more than one off-diagonal entry, triangle extension as described [11] is not possible on the arc (4, 1) in the digraph associated with A, demonstrating that Theorem 2.1 provides a more general technique than triangle extension in [11]. Example 2.5.…”

“…[1,5,13]). Recently in [11], a digraph method called triangle extension has been developed for constructing higher order spectrally or inertially arbitrary patterns from lower order patterns. In this paper, we generalize the triangle extension method by formulating it as a matrix bordering technique (see Remark 2.2).…”