Math 240 (J. Wu) Homework #7 due Friday March 5 (by 3pm)



READINGS:

  In Kreyszig, 7.9-7.10
  To be covered next week: 7.10-7.12


CORE PROBLEMS:

  Do all core problems in 7.9, 7.10
ADDITIONAL PROBLEMS:

  7.9  # 8,16,21
  7.10 # 20,24,25
  In # 20 you can use Maple to compute the characteristic polynomial and
  its  roots since this is a little messy by hand.
MAPLE ASSIGNMENT:
  1. You can again do this part of the assignment as a study group. But each person should attach a copy of the solutions to the assignment they hand in since otherwise the grading becomes too time consuming.
  2. Read about the commands in the Maple Manual that hepl you cpmpute eigenvalues and eigenvectors and look at the Demos on the Web.
  3. Do 7.10 # 18 and # 20 in Maple. "Take apart" the answers so you list the eigenvalues and eigenvectors in a form where they would be easy to use for other computations.


EXTRA CREDIT:

    [5 points]

    1. Show that for an nxn matrix A, the constant term in the characteristic polynomial is det(A) and that the coefficient of t(n-1) is, up to sign, trace(A) (which is defined to be the sum of the diagonal entries in A)
    2. Show that det(A) is the product of all the eigenvalues, counted with multiplicity, and that trace(A) is the sum of all the eigenvalues.