This the first time I deal with the matrix square rooting, there are many types of matrix decomposition techniques that factorize the matrix P into a form $P=SS^T$. For example, the Cholesky decomposition, the eigenvector decomposition or the singular value decomposition (SVD). I read that the most of mathematicians prefer to apply SVD, since it is the most robust algorithm to factorize a matrix especially when the matrix becomes nearly singular.
The first document I read it was a Phd thesis: The SVD of $P=UDU^T$, where U is matrix containing eigenvectors and D is a diagonal matrix of eigenvalues. The square-root of the matrix x is then given by $S=U\sqrt{D}U^T$
I want to know more about matrix square rooting using SVD: explanations and proofs, can someone give me some references or even if you can explain how it works by numerical examples in the solution? Thanks in advance.