Problem $3.[50$ points$]$ Use Jupyter notebook for parts $($b$),($d$)$ and $($e$)$ of this problem.

Consider the following matrix

$A=\left[\begin{array}{cc}5& -1\\ -1& -1\end{array}\right]$

Here you will write your own QR decomposition algorithm for a $22$ matrix and use it

repeatedly to find all the eigenvalues of $A($not just the largest eigenvalue$).$

a$.[10$ points$]$ Orthogonalize the column vectors of $A$ by hand. In other words, let $a_1=(5,-1)$

and $a_2=(-1,-1).$ Modify them using the QR decomposition method to find two orthogo$-$

nal vectors $e_1$ and $e_2.$ Show that two vectors $e_1$ and $e_2$ are indeed orthogonal. Write down

the matrix $Q=[e_1,e_2.].$

b$.[10$ points$]$ Write a Python function that takes an arbitrary $22$ matrix as input and returns

the matrix $Q.$ Provide $A$ to your function as an input and show that the returned $Q$ is equal

to the result from part $($a$).$

c$.[10$ points$]$ Construct the $R$ matrix in the QR decomposition by hand. Write down your

matrix $R.$

d$.[10$ points$]$ Add new code to the Python function defined in part $($b$),$ so that it returns $R,$

together with $Q.$ Show that your Python function now takes in $A$ as an input and returns $Q$

and $R.$

e$.[10$ points$]$ Apply the QR function your wrote in part $($d$)$ repeatedly to find two eigenvalues of

$A.$ Compare your numerical result with the eigenvalues found mathematically $($i$.$e$.,$ calculate

the eigenvalues by hand following the definition$).$