# The Solution of S exp(S) = A is Not Always the Lambert W Function of A

Corless, Robert M. and Ding, Hui and Higham, Nicholas J. and Jeffrey, David J. (2007) The Solution of S exp(S) = A is Not Always the Lambert W Function of A. [MIMS Preprint]

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We study the solutions of the matrix equation $S\exp(S) = A$. Our motivation comes from the study of systems of delay differential equations $y'(t) = A y(t-1)$, which occur in some models of practical interest, especially in mathematical biology. This paper concentrates on the distinction between \emph{evaluating a matrix function} and \emph{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.