I have to prove that
a matrix $A\in \mathbb{R}^{n \times n}$ is invertible if $A^T = p(A)$ where $p(A)$ is a polynomial with non-zero last coefficient.
I've tried to use that if $A^T=p(A)$ then $A=p(A^T)$ and to look at $$AA^T = (a_0E+a_1A^T+…+a_m(A^T)^m)(a_0E+a_1A+…+a_mA^m)$$ and somehow to get the expression like $cE=A(…)$ or $cE=A^T(…)$ which will prove what I need but failed.
Best Answer
If $A^T=p(A)$, then $A=p(A^T)=p(p(A))$. Since $p$ and therefore $p\circ p$ has a non-zero constant term, we can rearrange $A=p(p(A))$ to have the constant term (times the identity matrix) alone on one side, and a multiple of $A$ on the other side. Divide by the constant term, and you have something of the form $I=A\cdot q(A)$.
Alternately, $p(p(A))-A=0$ implies that $p(p(x))-x$ must be a multiple of the minimal polynomial of $A$. Since the minimal polynomial thus has a non-zero constant term, $A$ can't have $0$ as an eigenvalue.