Largest set of matrices that have unique root

Question

Does the set of all positive definite matrices is the largest set that one can get unique root?

for all matrices in the set $A \in \mathcal{P}(M_n(\mathbb C))$, for all n, the largest set that every matrix $B \in A$ have unique matrix $C \in A$ s.t $C^2=B$ is the set of all positive definite matrices?

and what if the matrix has unique nth root for all n ($C^n=B$)?

Answer 1

We can do better. Take $Z=\{B\in M_n(\mathbb{C})|spectrum(B)\subset T\}$, where $T=\{a+ib\in\mathbb{C}|b\geq 0$ and if $a\leq 0$, then $b>0\}$. Take $C=\exp(1/2\log(B))$ where $\log(.)$ is the principal logarithm.

EDIT. For the $n^{th}$ root, the answer is the same as above.