calculate high power of Matrix

Question

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how can I calculate $\pmatrix {1&0.5\\0&-0.5}^{2017}$ ?

For example, what are the methods to calculate the high power of the matrix

Answer 1

Let $A$ be your matrix. It has two eigenvectors, $(1,0)$, and $(-1,3)$, whose eigenvalues are $1$ and $-\frac12$ respectively. So, let$$T=\begin{pmatrix}1&-1\\0&3\end{pmatrix}.$$Then$$T^{-1}.A.T=\begin{pmatrix}1&0\\0&-\frac12\end{pmatrix}.$$Therefore,$$T^{-1}.A^n.T=\begin{pmatrix}1&0\\0&\left(-\frac12\right)^n\end{pmatrix},$$and so$$A^n=T.\begin{pmatrix}1&0\\0&\left(-\frac12\right)^n\end{pmatrix}.T^{-1}.$$