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Suppose the eigenvalues of a $3 \times 3$ matrix $A$ are $3,4 / 5,$ and $3 / 5,$ with corresponding eigenvectors $\left[\begin{array}{r}{1} \\ {0} \\ {-3}\end{array}\right],\left[\begin{array}{r}{2} \\ {1} \\ {-5}\end{array}\right],$ and $\left[\begin{array}{r}{-3} \\ {-3} \\ {7}\end{array}\right] .$ Let $\mathbf{x}_{0}=\left[\begin{array}{r}{-2} \\ {-5} \\ {3}\end{array}\right] .$ Find the solution of the equation $\mathbf{x}_{k+1}=A \mathbf{x}_{k}$ for the specified $\mathbf{x}_{0},$ and describe what happens as $k \rightarrow \infty$

Solution depends on the eigenvector decomposition. Since $\frac{4}{5}<1$ and $\frac{3}{5}<1$as $k$ increases solution $x_{k}$ looks more and more like its first part 2$\cdot(3)^{k} v_{1}$because $\left(\frac{4}{5}\right)^{k} v_{2} \rightarrow 0$ and $2 \cdot\left(\frac{3}{5}\right)^{k} v_{3} \rightarrow 0,$ as $k \rightarrow \infty$

Calculus 3

Chapter 5

Eigenvalues and Eigenvectors

Section 6

Discrete Dynamical Systems

Vectors

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