5. Let A be a symmetric, tridiagonal matrix. You learned that the matrices Ak defined by the QR-algorithm converge to a diagonal matrix that is similar to (and thus has the same eigenvalues as) A. The convergence speed depends on the absolute value of the ratio of consecutive eigenvalues. Let r ? (0, 1) and A = 1 r r 1 (a) Calculate the eigenvalues of A as a function of r (by hand). (b) Implement the QR-algorithm using MATLABs (or Pythons) implementation of the QR-factorization, qr(). Your code should run for a quadratic matrix of any size. (c) Now define a tolerance, e.g., ? = 10?10. Introduce a stopping criterion in your code, causing it to stop when the maximal difference between the true eigenvalues of A and the diagonal entries of Ak is smaller than ? . 2 (d) Use your code with the matrix given for at least five values of r ? (0, 1) and make a plot with r versus the number of iterations needed to achieve the given tolerance. Explain your findings by examining the ratio between the eigenvalues of A using (a).