18.06 Professor Johnson Quiz 3 May 1, 2009Grading1234Total:Your PRINTED name is:Please circle your recitation:(R01) M2 2-314 Qian Lin(R02) M3 2-314 Qian Lin(R03) T11 2-251 Martina Balagovic(R04) T11 2-229 Inna Zakharevich(R05) T12 2-251 Martina Balagovic(R06) T12 2-090 Ben Harris(R07) T1 2-284 Roman Bezrukavnikov(R08) T1 2-310 Nick Rozenblyum(R09) T2 2-284 Roman Bezrukavnikov1 (20 pts.) For each part, give as much information as possible about the eigen-values of the matrix A described in that part. (Each part describes adifferent matrix A. A may be complex.)(a) The recurrence uk+1= Aukhas a solution where kukk → 0 as k → ∞for one initial vector u0, but also has a solution with kukk → ∞ ask → ∞ for a different choice of the initial vector u0.(b) The equation (A2− 4I)x = b has no solution for some right-hand sideb.(c) A = eBTBfor some real matrix B with full column rank.(d) A = BTB for a 4×3 real matrix B, and the matrix BBThas eigenvaluesλ = 3, 2, 1, 0. (Hint: think about the SVD of B.)2This page intentionally blank.32 (20 pts.) You are given the matrixA =0.5 0.2 0.20.1 0.5 0.50.4 0.3 0.3.(i) What are the eigenvalues of A? [Hint: Very little calculation required!You should be able to see two eigenvalues by inspection of the formof A, and the third by an easy calculation. You shouldn’t need tocompute det(A − λI) unless you really want to do it the hard way.](ii) The vector u(t) solves the systemdudt= Aufor some initial condition u(0). If you are told that u(t) approachessome constant vector as t → ∞, give as much true information aspossible regarding the initial condition u(0).[Note: be sure you understand that this is not the same thing as solvingthe recurrence uk+1= Auk! Imagine how you would find u(t) if youknew what u(0) was.]4This page intentionally blank.53 (10 pts.) The 3 × 3 matrix A has three independent eigenvectors v1, v2, and v3withcorresponding eigenvalues λ1, λ2, and λ3(that is, Avi= λivifor i = 1, 2, 3).Ifb = c1v1+ c2v2+ c3v3for some coefficients c1, c2, and c3, then write (in terms of λi, ci, and vi) aformula for the solution x ofA2x + 2Ax − 3Ix = b(you can assume that a solution exists for any b).6This page intentionally blank.74 (15 pts.) A is a 3 × 3 real-symmetric matrix. Two of its eigenvalues are λ1= 1 andλ2= −1 with eigenvectors v1= (1, 1, 1) and v2= (1, −1, 0), respectively.The third eigenvalue is λ3= 0.(I) Give an eigenvector v3for the eigenvalue λ3. (Hint: what must betrue of v1, v2, and v3?)(II) Using your result from (I), write the matrix eAas the product of threematrices, and explicitly give the three matrices. (You need not workout the arithmetic, but your answer should contain no matrix inversesor matrix exponentials. If you find yourself doing a lot of arithmetic,you are forgetting a useful property of this matrix! )8This page intentionally
View Full Document
Unlocking...