$4\mathrm{.}25.$ Consider the vibrating system described by

$\left[\begin{array}{cc}4& 0\\ 0& 1\end{array}\right]x^{}(t)+\left[\begin{array}{cc}4& -2\\ -2& 2\end{array}\right]x(t)=0$

Compute the mass$-$normalized stiffness matrix, the eigenvalues, the normalized eigenvectors, the matrix $P,$ and show that $P^{T}MP=$I and $P^{T}KP$ is the diagonal matrix of eigenvalues $.$