A n × n-matrix A satisfies the equation A2 = A.
(a) List all possible characteristic polynomials of A.
(b) Show that A is similar to a diagonal matrix
The Attempt at a Solution
A2 = A
so, A2 - A = 0
A(A-I) = 0
Our minimal polynomial is x2 - x = m(x)
Our eigenvalues are 0 and 1, and since in our minimal polynomial each one has a multiplicity of 1, A is similar to a diagonal matrix consisting of n jordan blocks with either 1's or 0's on the diagonal and 0 everywhere else since we have an nxn matrix and n jordan blocks.
So I've proved part (b) first.
For part (a) we don't know how many times each eigenvalue occurs in the characteristic polynomial, so p(t) = ta(t-1)b
Is this correct?