Suppose I have a matrix $B := A + E$, where $A$ is diagonal and $E$ is an off-diagonal, symmetric matrix whose non-diagonal elements are small. Is there any way to obtain the approximate eigenvalue and eigenvector of $B$?
I came across this approximation
$$\lambda_B = \lambda_A + \frac{x^t E x}{x^t x}$$
where $x$ is the eigenvector of $A$. However, this does not work for me, because, $A$ is diagonal, and diagonal elements of $E = 0$. In this case, I get $x^t E x = 0$. Is there any other method?
Also assume the eigenvalues of $\lambda_A$ is positive, if it helps.