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.

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    $\begingroup$ Could you clarify, is C tridiagonal or triangular? I ask is it is the later that a Choleski decomposition produces $\endgroup$ – Ian Bush Nov 25 '18 at 10:54
  • $\begingroup$ Oh, I edit the original post. C is triangular $\endgroup$ – Kerem Nov 25 '18 at 11:56
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    $\begingroup$ The title asks about inverting the sum of identity and triangular matrices, which is tractable. But the body instead asks about an upper bound on the spectral norm of such an inverse. I don't think the problem statement lends itself to much more than a construction of the (triangular) inverse of a triangular matrix. $\endgroup$ – hardmath Nov 25 '18 at 21:20
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    $\begingroup$ There's no upper bound here- Take a single term with $C_{1}=\epsilon I$ the 2-norm of $(I+(C_{1}-I))^{-1}$ is $1/\epsilon$, which goes to infinity as $\epsilon \rightarrow 0$. $\endgroup$ – Brian Borchers Nov 26 '18 at 3:25
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    $\begingroup$ You can modify my earlier example to use $C_{1}=(1/2)I$ and $C_{2}=(1/2)I+\epsilon I$. $\endgroup$ – Brian Borchers Nov 26 '18 at 18:41

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