the primary value for matrix lambert w function

In this paper, we aim to find an initial solution for the matrix lambert w function, defined by w ew = a, where a; w 2 hn×n with q ⊆ h ⊆ c and n 2 n. our method utilizes only three key formulas within matlab to obtain this initial solution. the approach primarily leverages power series, along with considering similarities and differences from the derogatory case and canonical forms.