Integrators of matrices

A notion of differentiation for matrices through ‘differentiators’, where one is actually differentiating the characteristic polynomials, can be found in the literature. This concept is very useful in studying roots and critical points of complex polynomials. Here we introduce the inverse operation of integration for matrices through ‘integrators’. Surprisingly not all matrices are integrable in this sense. It is seen that non-derogatory matrices are best behaved with respect to this operation. They are ‘freely integrable’.