Consider the following matrix A:

$A=\left[\begin{array}{cccc}2\mathrm{.}9766& 0\mathrm{.}3945& 0\mathrm{.}4198& 1\mathrm{.}1159\\ 0\mathrm{.}3945& 2\mathrm{.}7328& -0\mathrm{.}3097& 0\mathrm{.}1129\\ 0\mathrm{.}4198& -0\mathrm{.}3097& 2\mathrm{.}5675& 0\mathrm{.}6079\\ 1\mathrm{.}1159& 0\mathrm{.}1129& 0\mathrm{.}6079& 1\mathrm{.}7231\end{array}\right]$

Design the following algorithms using the pseudocode presented in the class to determine the

eigenvalue$($s$)$ of $A.$

Algorithm $1$: Implement the Rayleigh Quotient Iteration to determine the eigenvalue of matrix

$A.$ Choose $4$ different starting vectors $e_1,e_2,e_3$ and $e_4.$ For each starting vector determine the

corresponding eigenvalue that the algorithm converges to$.$ Your code must produce the following

table as output. For convergence, use a tolerance of $0\mathrm{.}0001.$

Algorithm $2$: Design the QR Iteration algorithm to determine all the eigenvalues of $A.$ Note: For

the QR factorization, you can design the function gramSchmidt$()$ that employs the Gram$-$

Schmidt algorithm to get the $Q$ and $R$ matrices. The output of the algorithm should be:

The eigen values are:

The number of iterations for the convergence is:

Use a tolerance of $0\mathrm{.}0001$ between two successive iterations for every eigenvalue.

Note:

Use python to code and show results.

The algorithms must work for a general $nn$ matrix

You cannot use a built$-$in function to calculate either of the algorithms.

The program should print the table in Algorithm $1.$