I am trying to figure out whether the Cholesky decomposition is uniformly continuous. Specifically, I would like to derive a modulus of continuity for the Cholesky decomposition with respect to the spectral matrix norm, i.e., a monotonically increasing function $\omega: \mathbb{R}_+ \rightarrow \mathbb{R}_+$, such that
$\Vert\mathbf{L}_A-\mathbf{L}_B \Vert \leq \omega(\Vert\mathbf{A}-\mathbf{B} \Vert)$ holds for all symmetric positive definite matrices $\mathbf{A} = \mathbf{L}_A \mathbf{L}_A^{\top}$ and $\mathbf{B} = \mathbf{L}_B \mathbf{L}_B^{\top}$. Here $\Vert \cdot \Vert$ denotes the induced (spectral) norm $\Vert \mathbf{A} \Vert = \max_{\Vert\mathbf{x}\Vert=1} \Vert\mathbf{A}\mathbf{x} \Vert$.

I know that a similar bound holds for the positive-definite square root of a matrix, i.e., $\Vert \sqrt{\mathbf{A}} - \sqrt{\mathbf{B}} \Vert \leq C \Vert \mathbf{A} - \mathbf{B}\Vert^{\frac{1}{2}}$ for a constant C that potentially depends on the dimension of the matrices. I am also aware of [bounds on the derivative of the Cholesky decomposition mapping][1], which allows us to obtain bounds that depend on the condition number of the matrix $\mathbf{A}$. However, I require a bound that does not include the condition number, nor the norm of the inverse of $\mathbf{A}$ or $\mathbf{B}$
.

  [1]: https://d1wqtxts1xzle7.cloudfront.net/34000331/_Rajendra_Bhatia__Matrix_Analysis_%28Graduate_Texts_%28bookos-z1.org%29.pdf?1403311534=&response-content-disposition=inline%3B%20filename%3DGeneral_Theory.pdf&Expires=1618509996&Signature=BcF87ryMANpHSRGaVtqsqOPF2PDBZ~Xc1CNKvuk1kZ8UOtcRqq50IhtPk~d34mWOJPSsJAohrs4L~rzY5J0S~fyZjgshKmZDR2xXuhxpthLtCaAoAAk5qv5lV0TV5OqDRRPPaplVdT9GZjR93LokyrAx8bUK6VEKlCftUzhIwjCsxHG4kLbTDdknt8nLGKzWQNGEA2fUYiYVMbgr9-r~Of5USIbhLtvT35c7FqkGQGam~2Q-t-G2Lw3H9Kfc5HL6PlorHNeE49ZU8~y2Tl8e~yCqr0EY547KcnmlDBNnyxIjjSLXAkxoXaj3erBHvdwF8xGuHHoJlK8w9yEMLfuSHw__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA