(a) Find a matrix over with both minimal polynomial and characteristic polynomial equal to . Furthermore find two matrices and over which have the same characteristic polynomial, , and the same minimal polynomial, , but which are not conjugate to one another. Is it possible to find a third such matrix, , neither conjugate to nor to ? Justify your answer.
(b) Suppose is an matrix over which has minimal polynomial of the form for distinct roots in . Show that the vector space on which defines an endomorphism decomposes as a direct sum into , where is the identity.
[Hint: Express in terms of and
Now suppose that has minimal polynomial for distinct . By induction or otherwise show that
Use this last statement to prove that an arbitrary matrix is diagonalizable if and only if all roots of its minimal polynomial lie in and have multiplicity