Diagonalization & Matrix Powers · Challenge problem

Power iteration repeatedly multiplies a vector by a matrix $A$ and renormalizes; the error in the estimated dominant eigenvector shrinks by a factor of roughly $\left|\dfrac{\lambda_2}{\lambda_1}\right|$ every step, where $\lambda_1,\lambda_2$ are the two eigenvalues of $A$ with the largest absolute values. For a matrix with eigenvalues $6$ and $2$, find the smallest number of iterations $n$ for which $\left(\dfrac{2}{6}\right)^n < 0.01$.