Trace of the matrix power

Question

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!

Answer 1

You may do it by first computing matrix powers and then you may calculate whatever you want. Now question is how to calculate matrix power for a given matrix, say $A$? Your goal here is to develop a useful factorization $A = PDP^{-1}$, when $A$ is $n\times n$ matrix.The matrix $D$ is a diagonal matrix (i.e. entries off the main diagonal are all zeros). Then $A^k =PD^kP^{-1} $. $D^k$ is trivial to compute. Note that columns of $P$ are n linearly independent eigenvectors of $A$.

Answer 2

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*}