Finding the inverse of the eigenvectors matrix

Question

Hi I have the following $n\times n $ matrix: $$\left(\begin{array}{cccccc} 0 & 0 & 0 & 0 & & -0.5\\ 0.5 & 0 & 0 & 0 & & -0.5\\ 0 & 0.5 & 0 & 0 & 0 & \vdots\\ 0 & 0 & \ddots & 0 & 0 & -0.5\\ 0 & 0 & 0 & 0.5 & 0 & -0.5\\ 0 & 0 & 0 & 0 & 0.5 & -0.5 \end{array}\right) $$ I have managed to calculate its eigennvalues which are: $$\lambda_{k}=\frac{1}{2}exp\left({-\frac{2\pi ik}{n+1}}\right)$$ plus I found that the eigenvectors are: $$v_{k,j}=-\left(\sum_{j=1}^{m}\left(\frac{1}{2\lambda_{k}}\right)^{j}\right)=\frac{e^{\frac{2\pi ik}{n+1}}-e^{\frac{2\pi ik\left(m+1\right)}{n+1}}}{e^{\frac{2\pi ik}{n+1}}-1}$$ now i want to diagonalize this matrix, and to this I need to find $P^{-1}$ but I dont see a general way to invert the eigenvectors matrix, as $A$ is neither symmetric or normal, any help would be much appreciated.