Let M be a 2 \times 2 matrix with eigenvalues \lambda_1 = -1.2, \ \lambda_2 = 0.5 with corresponding eigenvectors \mathbf{v}_1 = \left[\begin{array}{c}
-2\cr
2
\end{array}\right], \mathbf{v}_2= \left[\begin{array}{c}
0\cr
1
\end{array}\right].
Consider the difference equation \mathbf{x}_{k+1} = M\mathbf{x}_k with initial condition \mathbf{x}_0 = \left[\begin{array}{c}
9\cr
5
\end{array}\right] .

Write the initial condition as a linear combination of the eigenvectors ofM .

That is, write\mathbf{x}_0 = c_1 \mathbf{v_1} + c_2 \mathbf{v_2}
= \mathbf{v_1} + \mathbf{v_2}

In general,\mathbf{x}_k =
\big( \big)^k \ \mathbf{v}_1+
\big( \big)^k \ \mathbf{v}_2

Specifically,\mathbf{x}_{2} = \left[\Rule{0pt}{2.4em}{0pt}\right. \left]\Rule{0pt}{2.4em}{0pt}\right.

For largek , \mathbf{x}_{k} \approx
\big( \big)^k\ \left[\Rule{0pt}{2.4em}{0pt}\right. \left]\Rule{0pt}{2.4em}{0pt}\right.

Write the initial condition as a linear combination of the eigenvectors of

That is, write

In general,

Specifically,

For large

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