Let M be a 3 \times 3 matrix with eigenvalues \lambda_1 = -1.2, \ \lambda_2 = 0.5 \ \lambda_3 = 2 with corresponding eigenvectors \mathbf{v}_1 = \left[\begin{array}{c}
2\cr
-2\cr
0
\end{array}\right], \mathbf{v}_2= \left[\begin{array}{c}
0\cr
-1\cr
-1
\end{array}\right], \mathbf{v}_3= \left[\begin{array}{c}
0\cr
1\cr
2
\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}
3\cr
3\cr
3
\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} + c_3 \mathbf{v_3}
= \mathbf{v_1} + \mathbf{v_2} + \mathbf{v_3}

In general,\mathbf{x}_k =
\big( \big)^k \ \mathbf{v}_1+
\big( \big)^k \ \mathbf{v}_2+
\big( \big)^k \ \mathbf{v}_3

Specifically,\mathbf{x}_{4} = \left[\Rule{0pt}{3.6em}{0pt}\right. \left]\Rule{0pt}{3.6em}{0pt}\right.

For largek , \mathbf{x}_{k} \approx
\big( \big)^k\ \left[\Rule{0pt}{3.6em}{0pt}\right. \left]\Rule{0pt}{3.6em}{0pt}\right.

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

That is, write

In general,

Specifically,

For large

You can earn partial credit on this problem.