I have stumbled upon a problem where I need the square root of the matrix
\begin{bmatrix} 0 & \cdots & 0 & x_1\\ \vdots & \ddots & \vdots & \vdots \\ 0 & \cdots & 0 & x_{n-1} \\ x_1 & \cdots & x_{n-1} & x_n \end{bmatrix}
It would be nice if the matrix square root wasn’t too complicated. Since this matrix has rank 2, I believe that a decomposition could indeed be quite simple. I am not able to verify this however, and would appreciate a little hint.