Consider a predator-pray model with the matrix $A= \begin{bmatrix} 0.5 & 0.6\\ -0.2 & 1.4 \end{bmatrix}$ ($A$ is the population distribution) and the initial state is $\vec{x}_0 = \begin{bmatrix} 1\\ 10 \end{bmatrix}$. Find the vector $\vec{x}_k$ for $k = 10$.
How can I find $\vec{x}_{k=10}$? And could I find $\vec{x}_{k=11}$ from $\vec{x}_{k=10}$?
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$\begingroup$The brute force way is simply to multiply it out; you need $A^{10}$, so calculate $A^2$, then $A^4$, then $A^8$, then multiply by $A^2$.
Another way is to find the eigenvalues and eigenvectors of the matrix. If you can diagonalize A, finding its powers is easy.
If $A = PDP^{-1},$ then $A^{10} = PD^{10}P^{-1}$.
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