The eigenvalues of matrices which commute with their derivative

Let F be a differential field whose field of constants is algebraically closed and let A be a matrix with coefficients in F which commutes with its derivative DA. We show that all the eigenvalues of A lie in F, answering open problem 22 of [1]. We also give a simple proof of a theorem of Schur characterizing matrices A with the property that the derivatives of A of all orders mutually commute. (C) 2013 Elsevier Inc. All rights reserved.