Exponential matrix equation I don't know how to solve (if it's possible) the following matrix equation:
$$\exp(H)=H^2,$$
where $H$ is a $N \times N$ hermitian matrix.
Does someone know if this equation has solutions and if the answer is yes, how to solve it?
Thanks in advance.
 A: *

*If you consider $P^{—1}HP$ where $P$ is invertible, and $H$ is a solution, then this matrix will be a solution. Since the matrix is supposed to be hermitian you can find an unitary matrix $P$ such that $P^*DP=H$, where $D$ is diagonal.

*Hence we deal the case $H$ diagonal. Each diagonal element satisfies the equation $e^x=x^2$, which is difficult to solve (we have to use special functions).

*The solutions are the matrices of the form $P^*DP$, where $P$ is unitary and $D=\operatorname{diag}(\lambda_1,\ldots,\lambda_N)$, with $\lambda_j^2=e^{\lambda_j}$ for all $j\in\{1,\ldots,N\}$.

A: I would like to give a different route to prove the conclusion of Davide's answer. As already stated, $H$ has a complete set of eigenfunctions and eigenvalues due to spectral theorem. So, given that $Hu_n=\lambda_n u_n$, we can consider the identity matrix written down as $I=\sum_n u_nu_n^\dagger$ and so, the initial identity just becomes
$$
  \sum_n(e^{\lambda_n}-\lambda_n^2)u_nu_n^\dagger=0
$$
and the equation for the eigenvalues immediately follows $e^{\lambda_n}-\lambda^2_n=0$. The solutions are given through Lambert W-function as already stated.
