I'm studying for an algebra exam, and this question was on a previous test:
Let $A$ be an invertible $n \times n$ complex matrix. Show that $A$ can be written as $$A = MN = NM$$ where $M$ is diagonalizable and all eigenvalues of $N$ are equal to 1.
I'm not really sure how to approach this sort of question; I know about Jordan normal form and rational canonical form and conceptually what eigenvalues are, but I don't have a sense of how to put all the information together to solve something like this. Any help would be appreciated!