If $A$ is normal, then so is any polynomial of it.

Question

By normal, I mean $AA^T$=$A^TA$

Just wondering if this statement is true or false. I'm trying to use this statement to prove something else in my homework. Thanks in advance.

Answer 1

Let $A$ be a normal matrix and $p$ a polynomial. Then $p(A)^T=p(A^T)$ since $(A^n)^T=(A^T)^n$ for any $n\in\Bbb N$ and $(A+B)^T=A^T+B^T$ for matrices $A$ and $B$. To show that $p(A)$ is normal, we need to show $p(A)p(A)^T=p(A)^Tp(A)$, so by the above we just have to show $p(A)p(A^T)=p(A^T)p(A)$. This is easy enough because if $x$ and $y$ commute then $p(x)$ and $p(y)$ commute.