The Lambert W function and the spectrum of some multidimensional time-delay systems

In this note we find an explicit expression for the eigenvalues of a retarded time-delay system with one delay, for the special case that the system matrices are simultaneously triangularizable  , which includes the case where they commute. Using matrix function definitions we define a matrix version of the Lambert WW function, from which we form the expression. We prove by counter-example that some expressions in other publications on Lambert WW for time-delay systems do not always hold.