EP548 Homework (not due), Spring 2002

Review of some important concepts from Chap 1-3: symmetry, positive definiteness, minimum principles
  1. What does it mean for a matrix A to be symmetric?
  2. If matrix A is a black box transforming input vector x into output vector y, how do you check whether A is symmetric?
  3. What is an inner product for vectors? for functions? for complex vectors? for complex functions?
  4. What does it mean for a linear differential operator to be symmetric? How do you check symmetry of a differential operator?
  5. Give mathematical examples of symmetric differential operators.
  6. Give physical examples that lead to symmetric differential operators.
  7. What is a Sturm-Liouville problem? what is special about them?
  8. Show that the 4th derivative operator is a symmetric operator for appropriate boundary conditions.
  9. Give physical examples that lead to a 4th order differential operator.
  10. What does it mean for a matrix A to be positive definite?
  11. When is a differential operator positive definite?
  12. Give examples of positive definite differential operators.
  13. Show that the 4th derivative operator with appropriate boundary conditions is positive definite.
  14. What can you say about the eigenvalues of a symmetric operator?
  15. What can you say about the eigenvectors of a symmetric matrix operator?
  16. What can you say about the eigenvectors of a symmetric differential operator?
  17. What is a skew-symmetric operator?
  18. What can you say about the eigenvalues and eigenvectors of a skew-symmetric operator?
  19. Let (x,y) denote the innner product of vectors x and y and x' denote the transpose conjugate of x. What is the minimum of F=(x,Ax)/2-(b,x) where A=[ 3, 2 ; 1, 5 ] and b=[1, 2]' ?
  20. For the same matrix A, what is the minimum of (x,Ax) subject to the constraint (x,x)=1?
  21. Find the general solution to -u''+ q u =0 where q is a given constant. Find q and u(x) so that u(0)=u(L)=0 for some positive L.
  22. Find the general solution to -u''+5u'-4u=0 and the specific solution that has u(0)=4 and u(1)=4e.
  23. Find a function u(x) in 0 < x < 3 such that u(0)=u(3)=0 that minimizes the integral from x=0 to 3 of [(u')2 - 2 u], where u' is du/dx.
  24. Find a function u(x) in 0 < x < 3 such that u(0)=u(3)=0 that minimizes the integral from x=0 to 3 of (u')2 subject to the constraint that the integral of u2=1.
  25. A solid body consists of a rectangular plate of length a and width b together with a rod of length c attached to a corner of the plate and perpendicular to it. The mass of the plate is M and the mass of the rod is m. Both are homogeneous. Find the center of gravity of the body. Find the principal moments and axes of inertia of the body. (For simplicity, assume that the plate and rod are infinitely thin).
  26. What is a companion matrix of the polynomial an xn + ... +a1 x + a0? Show that the eigenvalues of the companion matrix are the roots of the polynomial.
Strang 5.1
  1. Strang 5.1.1, 4, 6, 7, 12, 15.
  2. Derive the 9 point scheme for the Laplacian shown in exercise 5.1.5, what is its order of accuracy?
  3. Write pseudo-code to solve uxx + uyy = f(x,y) in the unit square with u=0 on the boundary using the method of steepest descent. Your code should not use any matrix explicitly.
  4. Write real code (in matlab preferably) to solve the above problem and compare your numerical solution to the analytic double sine series solution when f(x,y)=1.
  1. Strang 5.2.5, 9, 12, 18, 19, 21, 22
  2. Strang 5.3.8, 14, 15, 18, 19, 20, 23, 24