Assignment 8 – Jordan Forms and Applications, bis
Due April 3rd
i) A matrix M is called idempotent if M2 = M. Give an example of an
idempotent matrix that isn’t diagonal.
ii) Show that, if M is idempotent, then its eigenvalues must be 0 or 1.
iii) Show that M is idempotent if and only if M is similar to a diagonal matrix
which has only 0’s and 1’s on the diagonal and deduce that the Jordan
form of an idempotent matrix is determined by its size and trace.
iv) Verify that LeVerrier’s algorithm works for idempotent matrices.
Evaluation:
i) 1 pt, no partial credit
ii) 1 pt, no partial credit
iii) 3 pts, you may use ii) even if you didn’t prove it correctly
iv) 2 pts, you may use iii) even if you didn’t prove it correctly
