$n$th root of a matrix

Question

The following text is written in some paper I'd like to know how to define the map $\psi$ on the space of square (complex) matrices . As we know, for example, the square root of some real matrix is not necessary unique and if the matrix is diagonalisable (it is always diagonalisable in $\mathbb{C}$), we might right $$A = P^{-1}D P$$ and take the matrix $$B = P^{-1} E P,\quad E:= diag( \sqrt{D_{ii}})$$ to be the square root of $A.$
Is it the general definition of $\psi$? Taking into account the principle determination of $z\mapsto z^{\frac 1n}$ on $\mathbb{C}$ with $-\pi/n< arg(z)<\pi/n$ ?