I wonder if there are any theorems which can help me to calculate an upper bound for the spectral norm of: $$\left\| \left[ I + \sum_{i=1}^{\overline{n}\in\mathbb{N}} \big( C_i - I\big)\right]^{-1}\right\|_2$$ with the identity matrix $I \in \mathbb{R}^{n\times n}$ and $C_i$ are triangular matrices arising from a Cholesky decompostion of $A_i = C_i C_i^T $. The matrices $A_i \in \mathbb{R}^{n\times n}$ are symmetric positive semidefinite.
My first idea was to use that if $A\leq B$ the following relation holds for s.p.d. matrices $$\big\|(A + B)^{-1}\big\|_2 \leq \big\| A^{-1}\big\|_2.$$
But the problem is that the matrices $C_i - I$ are negative definite if I'm right.