= Integer powers for an annihilating polynomial with roots one and minus one
{title2=$(A-I)(A+I)^2=0$}
If a square <matrix> satisfies $(A-I)(A+I)^2=0$, then $A^{-1}=A^2+A-I$, and every integer <matrix power> lies in the span of $I,A,A^2$. Writing $s=(-1)^n$, the universal formula is $A^n=\alpha_nA^2+\beta_nA+\gamma_nI$, where $\alpha_n=(1-s)/4+ns/2$, $\beta_n=(1-s)/2$ and $\gamma_n=(1+3s)/4-ns/2$. The polynomial $p_n(t)$ on the right satisfies $p_n(1)=1$, $p_n(-1)=(-1)^n$ and $p_n'(-1)=n(-1)^{n-1}$. These value and derivative conditions implement interpolation at a repeated root. For negative integers the inverse formula first puts $A^n$ in the same polynomial algebra; the nilpotent block identity $(I-N)^n=I-nN$ for $N^2=0$ proves the derivative rule for every integer.
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