First-order perturbation of a simple symmetric eigenvalue ($\lambda (B+\epsilon E)=\lambda (B)+\epsilon x^{T}Ex+O({\epsilon }^2)$)

Let $B,E$ be fixed real symmetric matrices, with $Bx=\lambda x$, $\Vert x{\Vert }_2=1$, and $\lambda$ a simple eigenvalue. Complete $x$ to an orthonormal eigenbasis $x,x_2,\dots ,x_n$ with eigenvalues $\lambda ,{\lambda }_2,\dots ,{\lambda }_n$. Put $a=x^{T}Ex$ and $v=-{\sum }_{j=2}^n(x_j^{T}Ex)/({\lambda }_j-\lambda )x_j$. Then $(B+\epsilon E)(x+\epsilon v)-(\lambda +\epsilon a)(x+\epsilon v)={\epsilon }^2(E-aI)v$. The symmetric eigenvalue residual bound produces an eigenvalue $\lambda +\epsilon a+O({\epsilon }^2)$ in Big O notation.