Prof. Alan J. Laub December 11, 2014 EE 205A - FINAL EXAMINATION Fall 2014 Instructions: (a) The exam is closed-book (except for one two-sided page of notes) and will last 2 hours. Write your answers on your own paper; you may keep the exam itself. (b) You are encouraged to use a calculator although cell phones or similar Internetconnected electronic devices are NOT allowed. (c) Notation will conform as closely as possible to the standard notation used in the text Matrix Analysis for Scientists and Engineers and lectures. (d) Do all 5 problems; each is worth 20 points. Some partial credit may be assigned if warranted. Label clearly the problem number and the material you wish to be graded. 1. Let A = " 0 1 1 0 # and b = " # 1 . 1 (a) Find the unique solution of the inhomogeneous initial value problem x = Ax + b ; x(0) = x0 where the initial condition is x0 = " cos t sin t sin t cos t # " # 2 . You will find the fact that etA = 3 useful. (b) What can you say about the limiting solution x(t) as t ! +1. If the limit does not exist, is the solution bounded? 2. Let A 2 IRmn have an SVD A = UA A VAT with nonzero singular values 1 2 r pq T with nonzero singular . Similarly, let B 2 IR have an SVD B = U V r B B B s values 1 2 s . (a) What is the value of kA Bk2 where A B 2 IRmpnq . What is the value of kA BkF ? (b) What is a nearest rank-1 approximation to A B (expressed in terms of the components of the SVDs of A and B)? 3. (a) Let A 2 IRnn be symmetric and have eigenvalues 1 2 eA n. i. Show that must be symmetric. ii. What are its eigenvalues? iii. Under what conditions on the eigenvalues of A is the matrix eA positive definite? (b) Now let A 2 IRnn be skew-symmetric. i. Show that eA must be an othogonal matrix. ii. What are its eigenvalues? 4. Let A 2 IR44 be a matrix and suppose you know that 2 6 6 4 X=6 2 1 0 0 1 1 0 0 0 0 8 2 0 0 2 8 3 7 7 7 5 is a solution of the Lyapunov equation AX + XAT = explicitly, can you evaluate Z +1 I. Without solving for A etA dt ? 0 If yes, justify each step carefully. If no, why not? 5. (Master's Comprehensive Exam Question) Consider the second-order scalar linear dierence equation zk+2 = 4zk+1 " 4zk ; z0 = 2, z1 = 1. # zk 2 IR2 , write the scalar second-order dierence equation zk+1 as two first-order equations, i.e., find the 2 2 matrix A such that (a) Letting uk = uk+1 = Auk . (b) Find the initial condition u0 . (c) Find the general form of Ak by using the formula Ak = Xk X 1 where is the Jordan canonical form of A and X is a matrix of right eigenvectors (and right principal vectors if necessary) that reduces A to . (d) Solve the dierential equation, i.e., find uk = Ak u0 for k 0. (e) What is the unique solution of the second-order dierential equation, i.e., what is zk for k 0? 2