4.II.15C

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

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

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

*Typos? Please submit corrections to this page on GitHub.*