Spectrally Arbitrary Tree Sign Pattern Matrices

• Main Author: Kaphle, Krishna
• Format: Others
• Published: Digital Archive @ GSU 2006
• Subjects: Star pattern; Mathematics; Matrices; Tree pattern
• Online Access: http://digitalarchive.gsu.edu/math_theses/17; http://digitalarchive.gsu.edu/cgi/viewcontent.cgi?article=1016&context=math_theses

A sign pattern (matrix) is a matrix whose entries are from the set {+,–, 0}. A sign pattern matrix A is a spectrally arbitrary pattern if for every monic real polynomial p(x) of degree n there exists a real matrix B whose entries agree in sign with A such that the characteristic polynomial of B is p(x). All 3 × 3 SAP's, as well as tree sign patterns with star graphs that are SAP's, have already been characterized. We investigate tridiagonal sign patterns of order 4. All irreducible tridiagonal SAP's are identified. Necessary and sufficient conditions for an irreducible tridiagonal pattern to be an SAP are found. Some new techniques, such as innovative applications of Gröbner bases for demonstrating that a sign pattern is not potentially nilpotent, are introduced. Some properties of sign patterns that allow every possible inertia are established. Keywords: Sign pattern matrix, Spectrally arbitrary pattern (SAP), Inertially arbitrary pattern (IAP), Tree sign pattern (tsp), Potentially nilpotent pattern, Gröbner basis, Potentially stable pattern, Sign nonsingular, Sign singular