A4x4 matrix a with real entries and a4 = i, what are the possible characteristic polynomials of a

If , then the characteristic polynomial of is



which means has eigenvalues .

We know that if is an eigenvalue of , then is an eigenvalues of .

So if is the only eigenvalue of , we know that are the only possible eigenvalues of . We can construct five possible characteristic polynomials for in that case.