Past exam of the numerical linear algebra and continuous optimisation course of the University of Oxford / 2010 / A / 3 / d

For a unit eigenvector of $A$, $(A-I)x=(\lambda -1)x$. The matrix 2-norm bound from part (a) gives $|\lambda -1|\le 2/3$. Hence every eigenvalue lies inIn particular the condition number in the Euclidean operator norm satisfies ${\kappa }_2(A)={\lambda }_{\max }/{\lambda }_{\min }\le 5$.