On a symplectic vector space $(V,\omega)$ with an inner product $g$, one can construct a canonical almost complex structure using polar decomposition. If $(M,\omega)$ is a symplectic manifold with a Riemannian metric $g$, one can use this construction to construct an almost complex structure $J_p$ on each tangent space $T_pM$ of $M$ for all $p\in M$. Why is this construction smooth?
This problem is equivalent to the following matrix problem. Suppose $B$ is a positive definite symmetric matrix which depends smoothly on a parameter $t$ in some open interval $(a,b)$. Then there exists a unique positive definite symmetric matrix $S$ such that $S(t)^2 = B(t)$ for $t\in (a,b)$. Why does $S$ depend smoothly on $t$?