I have a symmetric positive definite covariance matrix $C$ and I'd like to get its square root $B$ such that $B'B=C$. My question is under what assumption(s) could this decomposition be unique, or unique up to something. Here's what I know, (1)If $B$ is triangular, then Cholesky decomposition could guarantee uniqueness. (2)If $B$ if symmetric positive definite, $B$ is unique as well. (3)Without further assumption, $B$ is generally not unique and is invariant to orthogonal transformation. Let $U$be unitary i.e. $U'U=I$ then any $UB$ satisfies $B'U'UB=B'IB=C$.
(4)If $B$ is symmetric(without PD assumption), $B$ is probably not unique. What do we know about this class of symmetric $B$?
(5)How to get a sparse $B$ (a lot of 0)?