\(\displaystyle \text{Let } A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix} \text{ and } P = \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}\)\( ,\theta>0 \). If \( B=PA P^{T},\; C=P^{T}B^{10}P \) and the sum of the diagonal elements of \( C \) is \( \frac{m}{n} \), where \( \gcd(m,n)=1 \), then \( m+n \) is:
