1. Let M,N be two commuting square matrices. Show that for each eigenvalue of M, there is an M-eigenvector (with that eigenvalue) that is also an N-eigenvector (with probably some other eigenvalue). Use this to show that there exists a basis in which M,N are both upper triangular, i.e. they're "simultaneously upper triangularizable".
2. Let M,N be two commuting diagonalizable matrices. Use last week's homework (perhaps) to show that they're simultaneously diagonalizable.
3. Let M = D1 + N1 = D2 + N2, where D1,D2 are diagonalizable, N1,N2 are nilpotent, and everybody commutes with everybody else. Show D1=D2.
4. Use JCF to show that if we feed a matrix into its characteristic polynomial, the result is the zero matrix. (It's very easy to show that it has all zero eigenvalues, but that's not good enough.)
5. Say M is in JCF, and p is its characteristic polynomial. If for each polynomial q of degree less than p, it is NOT true that q(M) = zero matrix, then describe M. (How big are the blocks, etc.)
6. Let M be a matrix as in #5. Show that T is a polynomial in M if, and only if, T commutes with M. (In fact the iff stuff goes deeper. If this statement is true about some M, then M is as in #5.)