= Solution
Because $A$ is a real symmetric <positive-definite matrix>, the <finite-dimensional spectral theorem> supplies an orthonormal eigenbasis $u_1,\ldots,u_n$ with
$$
0<\lambda_1\leq\cdots\leq\lambda_n.
$$
Its eigendecomposition is also its <singular value decomposition>. If $e=y^{(\delta)}-y$, then
$$
x^{(\delta)}-x=A^{-1}e
=\sum_{j=1}^n\frac{\langle e,u_j\rangle}{\lambda_j}u_j,
$$
so
$$
\boxed{\|x^{(\delta)}-x\|
\leq\frac{\delta}{\lambda_1}}.
$$
The operator norms satisfy $\|A\|_2=\lambda_n$ and $\|A^{-1}\|_2=1/\lambda_1$. Therefore the worst-case relative perturbation bound is
$$
\boxed{
\frac{\|x^{(\delta)}-x\|}{\|x\|}
\leq
\underbrace{\frac{\lambda_n}{\lambda_1}}_{\kappa_2(A)}
\frac{\|y^{(\delta)}-y\|}{\|y\|}}.
$$
The ratio $κ_2(A)$ is the <spectral condition number of a positive-definite matrix>. A large ratio means that data noise aligned with an <eigenvector> for the smallest <eigenvalue> is strongly amplified, so the inverse problem is ill conditioned.
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