I am trying to figure out how absolute errors of matrix elements in a real symmetric matrix propagate over to its eigenvalues.
My attempt:
Let $A$ be a real symmetric $n\times n$ matrix. Let $$\lambda_1 \geq \dots \geq \lambda_n$$ be its eigenvalues. We kan define the eigenvalues as functions of symmetric matrices, $\lambda_k(A) \quad \forall k=1,\dots, n$ since they are real and can be ordered. Assuming the eigenvalues are $C^1$, from Taylor's theorem one can make an error estimate for the eigenvalue $\lambda_k$ as function of the error of $A$: $$\left|\lambda_k(A+E) - \lambda_k(A)\right| \leq \sum_{i,j}\max_{c\in\langle0,1\rangle}\left|\frac{\partial\lambda_k}{\partial A_{ij}}(A+cE)\right|\left|\epsilon_{ij}\right| \leq \sum_{i,j}\max_{B}\left|\frac{\partial\lambda_k}{\partial A_{ij}}(B)\right|\left|\epsilon_{ij}\right|,$$ where $E=(\epsilon_{ij})_{ij}$ is a symmetric matrix of errors in $A$.
We have $\det(\lambda_k(A)I - A) = 0 \quad \forall \text{ real symmetric matrices }A$. Using the implicit function theorem and Jacobi's formula for derivative of determinant, I get$\DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\Co}{Co}$ $$\frac{\partial\lambda_k}{\partial A_{ij}}(A) = \frac{1}{\Tr\Co(\lambda_kI-A)}\Co (\lambda_kI-A)_{ij},$$ where $\Co$ stands for cofactor matrix.
For $n=2$ $$\frac{\partial\lambda_k}{\partial A}(A) = \frac{1}{(\lambda_k-a_{11}) + (\lambda_k-a_{22})} \left[\begin{matrix} \lambda_k - a_{22} & a_{21} \\ a_{12} & \lambda_k - a_{22} \end{matrix}\right],$$ and since $\det(\lambda_k(A)I - A) = 0, \quad (\lambda_k-a_{11})(\lambda_k-a_{22}) = {a_{12}}^2.$ Using this, we get $$\begin{aligned} \left|\frac{\partial\lambda_k}{\partial A_{11}}(A)\right| &= \left|\frac{\lambda_k - a_{22}}{(\lambda_k-a_{11}) + (\lambda_k-a_{22})}\right| \leq 1 \\ \left|\frac{\partial\lambda_k}{\partial A_{12}}(A)\right| &= \left|\frac{\sqrt{(\lambda_k - a_{11})(\lambda_k - a_{22})}}{(\lambda_k-a_{11}) + (\lambda_k-a_{22})}\right| \leq \frac{1}{2} \\ \left|\frac{\partial\lambda_k}{\partial A_{22}}(A)\right| &= \left|\frac{\lambda_k - a_{11}}{(\lambda_k-a_{11}) + (\lambda_k-a_{22})}\right| \leq 1 \end{aligned}.$$ This means $\left|\lambda_k(A+E) - \lambda_k(A)\right| \leq \left|\epsilon_{11}\right|+\left|\epsilon_{12}\right|+\left|\epsilon_{22}\right|$. The error propagates linearly with prefactor one for symmetric $2\times2$ matrices.
How can I generalize this result to real symmetric matrices of arbitrary dimension?