4. Gradient descent is a rstorder iterative optimisation algorithm for nding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the gradient of the function at the current point, because this is the direction of steepest descent. The iteration for the twodimensional case is given by the formula E-'1I1L+1 _ 1's. _ <..><.> We set E = ll]. Let ay) = 3.5:}:2 - 43y+y2 +1[l and In and ya he any real numbers. [a] Ebr all I, y E R, compute T'm. y} and nd a matrix A such that .(3-(;)-..... In) in terms of In and ya and powers of .4. ya [b] Find the eigenvalues of A. Write an expression for ( {c} Find one eigenvector corresponding to each eigenvalue. d Find matrices P and D such that D is di nal and A = PEP1. ago {e} Find matrices D", 131 and A". Find formulas for .1:n and 3;\". EN EN [g] Sketch the region R consisting of those [.rg, ya} such that my 3 G, Mr 2 and \\(\")l (\"4) HM Err1 where N is the number found in part {i}. Write an equation for the bounda.1:j"l.-r of R. \"nch points of the boundary belongs to R and which do not? {f} Suppose 11;. = ya = 1. Find the smallest N E N such that E 0.05. E G, ' ' L\": [1.05, F] [H P] [H H] F] H]