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The lambertw function satisfies

The lambertw(k, M) function computes the element-wise Lambert W function of M. The result, R, is formed as R[i,j] = lambertw(k, M[i,j]).

$\mathrm{with}\left(\mathrm{MTM}\right)\:$

$M≔\mathrm{Matrix}\left(2\,3\,'\mathrm{fill}'=3.5\right)\:$

$\mathrm{lambertw}\left(M\right)$

$\left[\begin{array}{ccc}1.130289327& 1.130289327& 1.130289327\\ 1.130289327& 1.130289327& 1.130289327\end{array}\right]$

$\mathrm{lambertw}\left(3\,M\right)$

$\left[\begin{array}{ccc}\mathrm{-1.595634197}+17.18618097\mathrm{I}& \mathrm{-1.595634197}+17.18618097\mathrm{I}& \mathrm{-1.595634197}+17.18618097\mathrm{I}\\ \mathrm{-1.595634197}+17.18618097\mathrm{I}& \mathrm{-1.595634197}+17.18618097\mathrm{I}& \mathrm{-1.595634197}+17.18618097\mathrm{I}\end{array}\right]$