Symmetric eigenvalue residual bound
= Symmetric eigenvalue residual bound
{title2=$\min_{\theta\in\operatorname{spec}(M)}|\theta-\mu|\leq\|Mu-\mu u\|_2/\|u\|_2$}
For a real <symmetric matrix> $M$, real $\mu$ and nonzero <vector> $u$, some <eigenvalue> of $M$ lies within the stated ratio of $\mu$. In an <orthonormal eigenbasis>, write $u=\sum_jc_jq_j$ and $Mq_j=\theta_jq_j$. Then $\|Mu-\mu u\|_2^2=\sum_j(\theta_j-\mu)^2c_j^2\geq\min_j|\theta_j-\mu|^2\|u\|_2^2$, proving the bound.