We study the solutions of the matrix equation S exp(S) = A. Our motivation comes from the study of systems of delay differential equations y ′ (t) = Ay(t − 1), which occur in some models of practical interest, especially in mathematical biology. This paper concentrates on the distinction between evaluating a matrix function and solving a matrix equation. In particular, it shows that the matrix Lambert W function evaluated at the matrix A does not represent all possible solutions of S exp(S) = A. These results can easily be extended to more general matrix equations.

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