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

Do all core problems in 7.9, 7.10

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.

- 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.
- Read about the commands in the Maple Manual that hepl you cpmpute eigenvalues and eigenvectors and look at the Demos on the Web.
- 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.

- 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) - 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.

[5 points]