Commutants of various matrices
It is a saddening fact about the universe in which we inhabit that not every linear operator is diagonalizable over 
. Meet the irritating matrix:
This matrix is guaranteed to haunt your nightmares and shatter whatever cozy and comforting illusions you may have about linear algebra. Some people refer to it as the Jordan block, but I would like to reserve the letter J for the matrix:
So that the irritating matrix is equal to 
.
The characteristic polynomial of the 
by irritating matrix is:
This is also the minimum polynomial. The only matrices which commute with the irritating matrix are those of the form:
In other words, a basis for the commutant of the irritating matrix is given by 
. Equivalently, a basis for the commutant of the irritating matrix is given by the first powers of the irritating matrix (starting from zero).
Luckily for us, the irritating matrix is as bad as things ever get. Over the complex numbers, any linear operator can be put into block diagonal form with irritating matrices corresponding to various 
along the diagonal. The operator is diagonalizable if the irritating blocks are all of size one.
The “best” kind of matrices are those which are not only diagonalizable, but also have distinct eigenvalues. For example:
For these matrices the characteristic polynomial is equal to the minimum polynomial and contains no repeated roots. In this case:
The only matrices which commute with such a matrix are other diagonal matrices. Equivalently, a basis for the space of linear operators
commuting with the original operator 
is given by 
To see the equivalence, note that to expand the diagonal matrix with eigehvalues 
in powers of the original matrix with eigenvalues 
you are looking for a polynomial
such that 
. Solving this linear system of equations involves inverting a vandermonde matrix, which is only possible if the eigenvalues are distinct.
If your operator is diagonalizable, but contains repeated eigenvalues, then the minimum polynomial still contains no repeated roots, but the characteristic polynomial does. For example:
Here the characteristic polynomial is: 
while the minimum polynomial is: 
.
The commutant of such a matrix is much larger than one with distinct eigenvalues. The commutant of our example matrix is:
where the stars denote possible nonzero entries.
Note finally, that both the irritating matrix and matrices with distinct eigenvalues have the property that we can find a vector 
such that the first powers of the matrix of form a basis. In the case of the irritating matrix take the vector:
In the case of the diagonal matrix with distinct eigenvalues take the vector:
This property can be used to characterize those matrices such that if 
commutes with then must be a polynomial in .