Invertible matrix is product of diagonalizable and matrix with eigenvalues 1

Question

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!

Answer 1

You know the Jordan normal form. Show that each Jordan block can be written in the required form, with $M=\lambda I$.

Finally conjugate the $M$ and $N$ block matrices back to the original coordinate system.

Answer 2

You know that $A$ is similar to an upper triangular matrix $T$. And an upper triangular matrix can be written as the product of a diagonal matrix $D$ (whose main diagonal consists of the elements of the main diagonal of $T$) and an upper triangular matrix $T^\star$ whose entries of the main diagonal are all equal to $1$.