TITLE: 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 [0 upvotes]: 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

REPLY [1 votes]: The statement is false. For example, any Jordan matrix with zeros on the diagonal satisfies the condition, but no such coefficients exist.
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