What are necessary and sufficient conditions that a matrix in $\mathbb R^{n\times n}$ must satisfy in order for its square root to exist, be symmetric, and also belong to $\mathbb R^{n\times n}$? If the square root is restricted to belong to $\mathbb R^{n\times n}\_+$ is it possible to guarantee uniqueness?

Sufficient conditions are simple to obtain, i.e. if the matrix is symmetric, invertible [**edit:** and has positive eigenvalues], but I have not been able to find necessary conditions.