New: 9.1.4, 9.1.6 Old: 8.1.4, 8.1.6. Find a complete set
of independent eigenvectors for each eigenvalue. Make sure to write
the null space for any multiple eigenvalues. No Gerschgorin. State
whether singular and defective.
New: 9.1.14 Old: 8.1.14 Find a complete set of independent
eigenvectors for each eigenvalue. Make sure to write the null space
for any multiple eigenvalues. No Gerschgorin. State whether
singular and defective.
New: 9.1.4, 9.1.6 Old: 8.1.4, 8.1.6. Redux. Check that is
indeed for the eigenvalues and eigenvectors you found. If
not, explain why not.
New: 9.2.11 Old: 8.2.13. First, verify that the book knows what
it is talking about by taking the matrix of 9.1.6/8.1.6,
and showing that can be diagonalized. Then check that can
indeed be diagonalized as the book says. Then prove that the
theorem is true for any arbitrary matrix . Bonus question
from the instructor: also prove, for any matrix , that if
is diagonalizable, is diagonalizable. Bonus qestion hint:
find eigenvectors and eigenvalues of in terms of those of .
New: 9.2.12 Old: 8.2.14. Show first that in the basis of the
eigenvectors,
To do so, show first that this is true for . Then show that if
it is true for a value , such as , it is true for the next
larger value of :
That will imply the desired result for
in
succession through recursion. Next show that since
Use the theorem to find a square root of the matrix of 9.2.5/8.2.5,
i.e. a matrix whose square is the matrix of question
9.2.5/8.2.5. Indicate .
New: 9.1.4 Old: 8.1.4. Redux. For the given matrix solve the
system
using the same method of
diagonalization as used in class. Draw typical solution curves in
the , plane.