1
$\begingroup$

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!

$\endgroup$
3
$\begingroup$

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.

$\endgroup$
0
$\begingroup$

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 $1$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.