Smigoc's glue for universal realizability in the left half-plane

TL;DR

This paper establishes sufficient conditions for the universal realizability of spectra in the left half-plane using companion matrices and migoc's procedure, including how adding negative real numbers can induce UR.

Contribution

It introduces a new sufficient condition for universal realizability of left half-plane spectra and characterizes families of UR lists, expanding understanding of spectral realizability.

Findings

Provides a sufficient condition for UR of left half-plane spectra.

Shows how adding a negative real number can make a non-UR list UR.

Characterizes families of UR left half-plane lists.

Abstract

A list of A list {\Lambda} of complex numbers is said to be realizable if it is the spectrum of a nonnegative matrix. {\Lambda} is said to be universally realizable (UR) if it is realizable for each possible Jordan canonical form allowed by {\Lambda}. In this paper, using companion matrices and applying a procedure by \v{S}migoc, is provides a sufficient condition for the universal realizability of left half-plane spectra. It is also shown how the effect of adding a negative real number to a not UR left half-plane list of complex numbers, makes the new list UR, and a family of left half-plane lists that are UR is characterized.