In the book "Computational Optimal Transport", the authors claim that the optimal transport map between two Multivariate Gaussian distributions $\mu$ and $\nu$ is given by: $$ T: x \mapsto m_\nu + A(x-m_\mu), $$ where $$ A = \Sigma_\mu^{-1/2} ( \Sigma_\mu^{1/2} \Sigma_\nu \Sigma_\mu^{1/2})^{1/2} \Sigma_\mu^{-1/2}= A^T $$
Now, I'm trying to calculate this matrix $A$ using a computer, and I was wondering if there is some Linear Algebra technique to do this computation efficiently. Does this matrix $A$ represents something known, like a decomposition or something?