Find a value b for which the power method on A+b*I is not converging

Question

I have approximated the largest and smallest eigenvalues of the matrix A by use of the power method. I am now asked to do something with these two values to come up with a value for b for which the power method on A+bI doesn't necessarily converge. I know that all eigenvalues of A will increase by b for A+bI but I am missing the next step. Does anyone have any tips?

Answer 1

If $\lambda_1$ and $\lambda_n$ are the largest and smallest eigenvalue of the matrix, then set $b:=\frac12(\lambda_1+\lambda_n)$. The matrix $$ A - \frac12(\lambda_1+\lambda_n) I $$ has smallest eigenvalue $- \frac12(\lambda_1-\lambda_n)$ and largest eigenvalue $ \frac12(\lambda_1-\lambda_n)$, i.e., they have the same absolute value. Now start with a linear combination of eigenvectors to these eigenvalues.