Trace of the matrix power Say I have matrix $A = \begin{bmatrix}
a & 0 & -c\\
0 & b & 0\\
-c & 0 & a 
\end{bmatrix}$.
What is matrix trace
tr(A^200)
Thanks much! 

For this kind of problems, first you have to get eigenvalues and their corresponding eigenvectors. For this problem,
$$ \lambda_1=b,\lambda_2=a-c,\lambda_3=a+c, v_1=(0,1,0)^T,v_2=(\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}})^T,v_3=(-\frac{1}{\sqrt{2}},0,\frac{1}{\sqrt{2}})^T. $$
Let 
$$ P=(v_1,v_2,v_3)=\left(\begin{matrix}0&\frac{1}{\sqrt{2}}&-\frac{1}{\sqrt{2}}\\
1&0&0\\
-\frac{1}{\sqrt{2}}&0&\frac{1}{\sqrt{2}}
\end{matrix}\right). $$
Then it is easy to check $P^T=P^{-1},P^TAP=\text{diag}(\lambda_1,\lambda_2,\lambda_3)$. Hence
\begin{eqnarray*}
\text{Tr}(A^{200})&=&\text{Tr}(P^TA^{200}P)\\
&=&\text{Tr}((P^TAP)^{200})\\
&=&\lambda_1^{200}+\lambda_2^{200}+\lambda_3^{200}\\
&=&b^{200}+(a-c)^{200}+(a+c)^{200}.
\end{eqnarray*}
