4. Gradient descent is a first-order iterative optimisation algorithm for finding 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 two-dimensional case is given by the formula In ) - EVf(In, Un). We set = = 0.1. Let f(x, y) = 3.5x - 4xy + 6.5y' + 10 and ro and yo be any real numbers. (a) For all x, y e R, compute Vf(x, y) and find a matrix A such that [3] A (?) = (7 -EV1(2,3). Write an expression for I'm ) in terms of to and yo and powers of A. (b) Find the eigenvalues of A. [1] (c) Find one eigenvector corresponding to each eigenvalue. (2] (d) Find matrices P and D such that D is diagonal and A = PDP-1. [1] (e) Find matrices D", P- and A". Find formulas for a, and yn. [4] (f) Suppose To = 30 = 1. Find the smallest NEN such that (IN ) 5 0.05. [3] (g) Sketch the region R consisting of those (ro, yo) such that IN 2 0, yx 2 0 and [4] I (IN) | 0.05. (2X-:)10.05. where N is the number found in part (f). Write an equation for the boundary of R. Which points of the boundary belongs to R and which do not