$[$QR$-$algorithm, $4+2+3+6pts$ Let $A$ be a symmetric, tridiagonal matrix. We know that the

matrices $A_k$ 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 rin$(0,1)$ and

$A=\left[\begin{array}{cc}1& r\\ r& 1\end{array}\right]$

$($a$)$ Calculate the eigenvalues of $A$ as a function of $r($by hand$).$

$($b$)$ Implement the QR$-$algorithm using MATLAB's $($or Python's$)$ implementation of the QR$-$

factorization, $).$ Your code should run for a square matrix of any size.

$($c$)$ Now define a tolerance $={10}^{-10}.$ 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 $A_k$ is smaller than $.{?}^1$

$($d$)$ Use your code with the matrix given for at least five values of rin$(0,1)$ 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$).$