# 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

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 $$$$a_{2n-1}J^{2n-1}+a_{2n-2}J^{2n-2}+\cdots+a_1 J=I_n.$$$$

Any help is appreciated

• Thanks. What if $J$ has no zero eigenvalues? – James Vestal May 23 at 22:00
• Did you notice that J is $n$ by $n$ but we have $J^{2n-1}$ present? – James Vestal May 23 at 22:26
• @JamesVestal Yes, and I don't see how that makes a difference. We can simply take $a_{n+1} = a_{n+2} = \cdots = a_{2n - 1} = 0$, and apply the process described in the answers to the post to find the remaining coefficients. – Ben Grossmann May 23 at 22:30