I faced an algebraic equation that I'm not sure if there's a closed form solution for it. The equation is $$ \bf{A} Z + b \exp(\mu + 1^T.Z) = 0 $$ Where $\bf{A}$ is a diagonal matrix and $\bf{b}, 1, Z$ are vectors.
I know that when $z$ is scalar, there's a closed-form solution for such equations. $$ z = -W\left(a b \exp(\mu)\right) $$ Where $W$ is the Lambert-W function.
Does such solution exist when $\bf{Z}$ is a vector? Or I should use numerical methods to find the roots of this system?
Assuming $\bf A$ is invertible, multiply your equation by ${\bf A}^{-1}$ on the left to see that ${\bf Z} = z {\bf A}^{-1} {\bf b}$ where $$ z = - \exp(\mu + {\bf 1}^T {\bf Z}) = -\exp(\mu + z \; {\bf 1}^T{\bf A}^{-1} {\bf b})$$ This is now a scalar equation where you can use LambertW.