Implement power iteration and verify the convergence rate

Implement power iteration for a symmetric matrix $A$: starting from a random unit vector $v_0$, repeat $v_{t+1} = Av_t/\|Av_t\|_2$ and estimate $\lambda_{\max}$ by the Rayleigh quotient $v_t^\top A v_t$.

1. Test it on a diagonal matrix with known spectrum.
2. Measure the error $|\lambda_t - \lambda_1|$ against $t$ on a log scale and verify that it decays geometrically with ratio $(\lambda_2/\lambda_1)^2$.
3. Construct a matrix with $\lambda_1 = \lambda_2$ and explain what happens.