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.
|Journal||Journal of Mathematical Analysis and Applications|
|State||Published - 15 Jun 2021|
Bibliographical noteFunding Information:
Investigations of integrability for diagonalizable matrices ( Theorem 3.13 ) are supported by the Ministry of Science and Higher Education of the Russian Federation (Goszadaniye No. 075-00337-20-03, project No. 0714-2020-0005 ). Necessary and sufficient conditions for a matrix integral to be diagonalizable ( Theorem 4.1 ) are obtained under the financial support of the Russian Federation Government (Grant number 075-15-2019-1926 ). The dual version of the Schoenberg inequality ( Theorem 5.1 ) was obtained under the financial support of RGC grants 17301115 and 17307420 .
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