Consider A=(2003)A = \begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}A=(2003) and Aϵ=(2ϵϵ3)A_\epsilon = \begin{pmatrix} 2 & \epsilon \\ \epsilon & 3 \end{pmatrix}Aϵ=(2ϵϵ3) for small ϵ>0\epsilon > 0ϵ>0. Which describes the eigenvalues of AϵA_\epsilonAϵ?
They remain exactly 2 and 3 for all ϵ\epsilonϵ
They shift continuously from 2 and 3, staying real
They become complex for any ϵ>0\epsilon > 0ϵ>0
One eigenvalue stays fixed at 2.5