8 Prove or give a counterexample If A and B have the same ch

8. Prove or give a counterexample: If A and B have the same charac teristic polynomial, then A and B are similar.

Solution

Suppose A and B are n x n matrices and A is nonsingular.

Then,

BA = A ^-1 (AB)A

and so AB and BA are similar.

Thus, they have the same characteristic polynomial, and the same eigenvalues and multiplicities. The same argument applies if B is nonsingular, using

AB = B ^-1 (BA)B.

However, if A and B are both singular, then AB and BA need not be similar.