Is it true that $(A^{1/2})^T = (A^T)^{1/2}$ for positive definite matrix A

Question

I'm currently working on a proof in which the term $(A^{1/2})^T$ for some positive definite matrix A $(m \times n)$ is present. Manipulating the term into $(A^{T})^{1/2}$ would greatly benefit me, but is it generally true? A simple reasoning shows \begin{equation} A^{1/2} = B \iff A = BB \end{equation} and thus \begin{equation} A^{1/2} = B \implies (A^{1/2})^{T} = B^T \end{equation} \begin{equation} A^T = (BB)^T = B^TB^T \implies (A^T)^{1/2} = B^T \end{equation} But is this enough to establish that $(A^{1/2})^T = (A^T)^{1/2}$ ?