I'm trying to prove the following statement.  Let $P_n \subset Mat_{nxn}(\mathbb R)$ be the set of all symmetric positive definite matrices with real entries of size $n$x$n$.  Let $\sqrt{}:P_n \to P_n$ such that $\sqrt{A}=B$ where $B^2=A$ (From Linear Algebra 2, we know that there is such a matrix and there is only one, so this function is well defined).  Show that $\sqrt{}$ is smooth $C^{\infty}$, meaning, it is differentiable $\infty$ times and each differential (which can be seen as a linear transformation) is continuous.  Would appreciate any push in the right direction. This is not homework, I am preparing for an exam.
Let $M$ be the manifold of real positive definite $n \times n$ matrices, define a mapping $i:A \to \sqrt A$ (where $A\in M$ and $\sqrt A$ means the unique positive definite square root of $A$). Please show that $i$ is smooth.
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