If A is a symmetric matrix and a numerically stable algorithm is used to compute the eigenvalues

Question:

If A is a symmetric matrix and a numerically stable algorithm is used to compute the eigenvalues of Ax = b, then the relative error in the computed eigenvalues should always be small.
In the case of a true statement, explain or prove your answer. In the case of a false statement, give an example to show that the statement is not always true.