Let $K(x)\_{n\times n}$ be a positive definite matrix defined on $x\in D$ and $K\_{i,j}(x)\in C^2(D)$ (or generally $C^k$) for any $1\le i,j\le n$. Of course for any $x$, there exists a invertable matrix $A(x)$ such that $A(x)A(x)^t=K(x)$, and $A(x)$ is not unique. What I want to ask is, whether $K(x)$ has $C^2$ (or $C^k$) decomposition. That is, there exists $A(x)\in C^2(D)$ (or $C^k$), such that $A(x)A(x)^t=K(x)$. If not, what conditions can we add to make this true?


Moreover, are there some references about this question? Thanks a lot!



Decomposition of a positive definite matrix
The [Cholesky decomposition](https://en.wikipedia.org/wiki/Cholesky_decomposition) does what you want. It depends smoothly on the input matrix, because every step in the algorithm is a smooth function. It's all just basic arithmetic and square roots.


