Integrability of diagonalizable matrices and a dual Schoenberg type inequality

TL;DR

This paper investigates the integrability of diagonalizable matrices, characterizes it via spectral properties, and applies findings to establish a dual Schoenberg type inequality relating polynomial zeros and critical points.

Contribution

It provides a spectral characterization of integrability for diagonalizable matrices and connects this to polynomial zero-critical point inequalities.

Findings

Characterization of integrability in terms of spectrum

Conditions for the diagonalizability of the integral

A dual Schoenberg type inequality for polynomial zeros and critical points

Abstract

The concepts of differentiation and integration for matrices were introduced for studying zeros and critical points of complex polynomials. Any matrix is differentiable, however not all matrices are integrable. The purpose of this paper is to investigate the integrability property and characterize it within the class of diagonalizable matrices. In order to do this we study the relation between the spectrum of a diagonalizable matrix and its integrability and the diagonalizability of the integral. Finally, we apply our results to obtain a dual Schoenberg type inequality relating zeros of polynomials with their critical points.