E231 HW 4, due Friday, October 14, 5:00 PM1. Suppose v ∈ Cn. Show thatkvk∞≤ kvk2≤ kvk1Also, find a vector v such that all of the norms are equal, showing that the in-equalities are “tight”.2. Let A ∈ Cn×n. For k = 1, 2, . . ., define Nk:= NullSpaceAk.(a) Show that for all k, Nk⊂ Nk+1.(b) If for some integer p, Np= Np+1, then Np= Nqfor all q ≥ p.(c) Do the above results hold for general linear operators A ∈ L(V, V )?3. Consider a linear operator A ∈ B (C5, C4) defined by matrix multiplication: forv ∈ C5, A(v) = Av, where A isA =1 2 1 1 −1−1 −2 1 2 11 1 3 4 11 1 1 1 1(a) Find basis sets for Range(A), Range(A ◦ A∗), Ker(A∗), Ker(A∗◦ A). Notethese are all subspaces of C4.(b) Find basis sets for Range(A∗), Range(A∗◦ A), Ker(A), Ker(A ◦ A∗). Notethese are all subspaces of C5.(c) Verify that all of the orthogonality and direct sum properties hold amongthese subspaces.(d) With respect to the basis you chose for Range(A) and Range(A∗), find thematrix representation of the operatorA|Range(A∗)→ Range(A)Verify that this is an invertible operator (recall that it always is).4. Using the data from problem 3, find the orthogonal projection onto Range(A).5. Using the data from problem 3, characterize and solve the least squares problemfor Ax = b withb =1−4−30, and b =100016. A particle of mass m(= 2) is lying on a frictionless table. We apply a force u(t)at time t. Suppose at t = 0, the particle is at rest, and (with respect to somereference on the table) the position is 0. Let d and v be the position and velocityof the particle.(a) Show that for all t ≥ 0"d(t)v(t)#=Zt0"1m(t − τ)1m#u(τ)dτ(b) Suppose we want to choose u so that at t = 1, the position and velocitysatsify d(1) = 2, v(1) = −1. Find the minimum norm force u that achievesthis “transfer”, withkuk2=Z10u2(t)dtPlot the force, displacement and velocity as functions of time.(c) How would you modify your approach if the initial position and velocity (att = 0) were nonzero? What would the operator and its adjoint be?7. Find a matrix A ∈ R2×2, with positive, real eigenvalues, such that for some nonzerovector x, xTAx < 0. Can such an A be symmetric?8. Suppose F is either R or C. Using the matrix inversion lemma, and Schur com-plements, show the following: Given X ∈ Fn×n, Y ∈ Fn×n, with X = X∗> 0, Y =Y∗> 0, and a positive integer r. Show that there exist matrices X2∈ Fn×r, X3∈Fr×rsuch that X3= X∗3and"X X2X∗2X3#> 0 ,"X X2X∗2X3#−1="Y ?? ?#if and only if"X InInY#≥ 0 , rankX − Y−1≤ r.9. Suppose A ∈ Cn×nis Hermitian and positive definite, so A = A∗> 0. Show thathx, yiA:= x∗Ayis an inner product on Cn. How/Why is the assumption of Hermitian important?How/Why is the assumption of positive-definiteness important?10. The following is a singular value decomposition of A ∈ R3×2A =35−450453500 0 13 00 0.10 0"12−√32√3212#2(a) Is ATA an invertible matrix? Is AATan invertible matrix?(b) What is an eigenvalue/eigenvector decomposition of AAT?(c) What is an orthonormal basis for the null space of AT?(d) What is an orthonormal basis for the range space of A?(e) What is an orthonormal basis for the null space of A?(f) What is kAk2,211. Suppose X, Y ∈ Cn×n, with X = X∗> 0, and Y = Y∗≥ 0. Show thatmaxx∈Cnkxk2=1x∗Y xx∗X−1x= maxx∈Cnx6=0nx∗Y xx∗X−1x= λmaxX1/2Y X1/2= ρ (XY )What if C is replaced by R, everywhere?12. Suppose that W ∈ Cn×n, with W = W∗, and L ∈ Cn×nis invertible. Show thatW < 0 if and only if L∗W L < 0. Does the result hold when < is replaced by oneof >, ≤, ≥?13. Find, by hand calculation the eigenvalues and the eigenvectors for the matrices(a) A ="5 −412 −9#(b) A ="14 −824 −14#(c) A ="0 1−9 −4.8#(d) A ="−2.4 1.8−1.8
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