Linear Mathematics | Part IB, 2001

Let AA and BB be n×nn \times n matrices over C\mathbb{C}. Show that ABA B and BAB A have the same characteristic polynomial. [Hint: Look at det(CBCxC)\operatorname{det}(C B C-x C) for C=A+yIC=A+y I, where xx and yy are scalar variables.]

Show by example that ABA B and BAB A need not have the same minimal polynomial.

Suppose that ABA B is diagonalizable, and let p(x)p(x) be its minimal polynomial. Show that the minimal polynomial of BAB A must divide xp(x)x p(x). Using this and the first part of the question prove that (AB)2(A B)^{2} and (BA)2(B A)^{2} are conjugate.

Typos? Please submit corrections to this page on GitHub.