  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.
