Uniqueness of the square-root of the diffusion matrix?

In the Langevin equation with hydrodynamic interactions the stochastic force on particle $a$ is: $$\sqrt{2k_BT} A^{ab}_{ij} \xi^{b}_j(t)$$ where $\xi$ is a unit white noise. Here $A^{ab}_{ij}$ is the square root of the mobility matrix (and spare a multiplicative constant the diffusion matrix) in the sense that: $$M_{ij}^{ab}=A_{ik}^{ac}A^{bc}_{jk}\tag{1}$$ My question is: does (1) uniquely define $A$ and if not how do we choose the 'correct' $A$?

That being said there are some typical ways in which the matrix $A$ (in the literature more commonly called $B$) is found. These include things like Cholesky decomposition.