Consider the following matrix

A $=$

$51$

$11$

Here you will write your own QR decomposition algorithm for a $2\backslash$times $2$ 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 orthogonal 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 $2\backslash$times $2$ 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$).$