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.

中文翻译：

---

对角化矩阵的可积性与对偶Schoenberg型不等式

为了研究复多项式的零点和临界点，引入了矩阵的微分和积分概念。任何矩阵都是可微的，但是并非所有矩阵都是可积的。本文的目的是研究可积性，并在对角化矩阵类中对其进行表征。为了做到这一点，我们研究了对角化矩阵的频谱与其可积性和积分的对角化性之间的关系。最后，我们应用我们的结果来获得对偶多项式零与临界点相关的对偶Schoenberg型不等式。