Conditions for a matrix to commute with its integral
Irving J. Epstein · Proceedings of the American Mathematical Society · 1963
to hold in an interval 0< t < to, where to is so small that throughout the interval [0, to] the Jordan canonical form of U(t) has the same form. That is, its off-diagonal elements do not change in the interval. Matrices U(t) satisfying (1.1) are of interest for various reasons; see, for instance, [1, p. 278]. We may mention two occasions where (1.1) occurs. Firstly, consider a system of n homogeneous linear differential equations of the first order for n unknown functions with U(t) as the matrix of coefficients. If we consider the unknown functions as components of a vector, and if we form a matrix Y, the n columns of which are n linearly independent solutions of our system, then we have for Y= Y(t):