Let A and B be nn matrices Prove that the matrix products AB

Let A and B be n×n matrices. Prove that the matrix products AB and BA have the same eigenvalues.

Solution

If ABv = v, then BAw = w, where w = Bv,

Thus, as long as w not equals to 0, it is an eigenvector of BA with eigenvalue .

However, if w = 0, then ABv = 0, and so the eigenvalue is = 0, which implies that AB is singular,But then so is BA, which also has 0 as an eigenvalue.

Thus every eigenvalue of AB is an eigenvalue of BA.