I am trying to figure out whether the Cholesky decomposition is uniformly continuous within a set of matrices with bounded entries. 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 $\mathbf{B} = \mathbf{L}_B \mathbf{L}_B^{\top}$ with $[\mathbf{A}]_{ij}, [\mathbf{B}]_{ij} \leq C$ for all $i,j \leq n$ and some $0< C < \infty$. 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 [1]. I am also aware of bounds on the derivative of the Cholesky decomposition mapping [2], 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] - Rajendra Bhatia, Matrix Analysis, Theorem X.l.l
[2] - Rajendra Bhatia, Matrix Analysis, Problem X.5.9
