Let $n$ be a natural integer with $n \geqslant 2$. For any real number $x$, we consider the following matrix in $\mathscr{M}_{n}(\mathbb{R})$
$$M_{x} = \left(\begin{array}{ccccc} x & 1 & \cdots & 1 & 1 \\ 1 & x & \cdots & 1 & 1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \cdots & x & 1 \\ 1 & 1 & \cdots & 1 & x \end{array}\right)$$
Show that the matrix $-M_{0}$ is diagonalizable and determine its eigenvalues and eigenspaces.