= First-order perturbation of a simple symmetric eigenvalue
{title2=$\lambda(B+\varepsilon E)=\lambda(B)+\varepsilon x^TEx+O(\varepsilon^2)$}
Let $B,E$ be fixed real <symmetric matrices>, with $Bx=\lambda x$, $\|x\|_2=1$, and $\lambda$ a <simple eigenvalue>. Complete $x$ to an <orthonormal eigenbasis> $x,x_2,\ldots,x_n$ with <eigenvalues> $\lambda,\lambda_2,\ldots,\lambda_n$. Put $a=x^TEx$ and $v=-\sum_{j=2}^n(x_j^TEx)/(\lambda_j-\lambda)\,x_j$. Then $(B+\varepsilon E)(x+\varepsilon v)-(\lambda+\varepsilon a)(x+\varepsilon v)=\varepsilon^2(E-aI)v$. The <symmetric eigenvalue residual bound> produces an <eigenvalue> $\lambda+\varepsilon a+O(\varepsilon^2)$ in <Big O notation>.
Back to article page
