4. Let 1.0] 0.99 A: one one] [a] Find the eigenvalue decomposition of A by hand. Recall that A is an eigenvalue of A if for some u[1], a[2] {entries of the corresponding eigenvector] we have {1.01 A}a[1] + ago-up] = n .99u[1] + (one A}u[2] = o. Another way of saying this is that we want the values of A such that A M (where I is the 2 x 2 identity matrix} has a nontrivial null space there is a nonzero vector n such that (A Mm = I]. Yet another way of saying this is that we want the values of A such that det{A AI} = ll Once you have found the two eigenvalues, you can solve the 2 x 2 systems of equations Aul = Ayn] and Aug = A2152 for 11.1 and Hg. Show your work above, but feel free to check you answer using MATLAB ,u'numpy. {b} If y = [1 1]T, determine the solution to An: = y. (:3) Now let y = [1.1 1] T and solve An; = y. Comment on how the solution changed. {d} Suppose we observe y = A1: + c with ||e||2 = 1. We form an estimate 5: = 2113;. Which vector 9 [over all error vectors with He\"; = 1) yields the maximum error \"3': mug? (e) Which {unit} vector 3 yields the minimum error? {1'} Suppose the components of e are independent and identically distributed (i.i.d.) Gaussian random variables: 3 m Normal, I}. 1What is the meansquare error IE1[||:i3 mug]