Let S belong to F(m x n) and T belong to (n x m). Let ST be invertible. Prove that ST is diagonalizable if and only if TS is diagonalizable for: a) when n = m b) when n > m c) Can ST be invertible when n < m?
I tried a lot in different ways but couldn't get this done. Can anyone help me on this? Thanks in advance. :)
Hints for (b). Denote the minimal polynomials of $ST$ and $TS$ by $m_{ST}$ and $m_{TS}$ respectively.
Suppose $ST$ is diagonalisable. Since $ST$ is invertible, $0$ is not a root of $m_{ST}$. Hence $p(x)=x\,m_{ST}(x)$ is a product of distinct linear factors. Now, show that $p(TS)=0$ and hence $m_{TS}|p$.
Conversely, suppose $TS$ is diagonalisable, so that $m_{TS}$ is a product of distinct linear factors. By considering $S\,m_{TS}(TS)\,T$ and by using the assumption that $ST$ is invertible, show that $m_{TS}(ST)=0$ and hence $m_{ST}|m_{TS}$.