Show that there exist $a_1,\ldots, a_{2n-1}$ such that $ a_{2n-1}J^{2n-1}+\cdots+a_1 J=I_n,$ where $J$ is a Jordan matrix

Question

Let $J\in\mathbb{C}^{n\times n}$ be a Jordan normal form and assume that ${\rm tr~}J<2n$. Prove or disprove that there exist $a_1,\ldots, a_{2n-1}\in\mathbb{R}$ such that \begin{equation} a_{2n-1}J^{2n-1}+a_{2n-2}J^{2n-2}+\cdots+a_1 J=I_n. \end{equation}

Any help is appreciated

Answer 1

The statement is false. For example, any Jordan matrix with zeros on the diagonal satisfies the condition, but no such coefficients exist.