91Ó°ÊÓ

A function is defined as a relationship between a set of inputs that each have one output.

Given,

$$\begin{gathered} A= \left[2         2        1 \\ 0         0       0 \\ 1          1         2\right],\stackrel{→}{x_0}=\left[0 \\ 2 \\ 0\right] \end{gathered}$$

Solve,

$\det \left(A- \mathrm{λ}I\right)=0$

$$\begin{gathered} \left|2-\mathrm{λ}         2                   1 \\ 0            -\mathrm{λ}           0 \\ 1                  1               2-\mathrm{λ}\right|=0 \\ -\mathrm{λ}\left({\left(2-\mathrm{λ}\right)}^2-1\right)=0 \\ -\mathrm{λ}\left({\mathrm{λ}}^2-4\mathrm{λ}+3\right)=0 \end{gathered}$$

$$\begin{gathered} -\mathrm{λ}\left(\mathrm{λ}-1\right)\left(\mathrm{λ}-3\right)=0 \\ {\mathrm{λ}}_1=0, \\ {\mathrm{λ}}_2=1, \\ {\mathrm{λ}}_3=3 \end{gathered}$$

Solve for $ \mathrm{λ}=0$

$$\begin{gathered} Ax=0 \\ \left[2            2            1 \\ 0           0            0 \\ 1             1              2\right]  \left[x_1 \\  x_2 \\ {x}_3\right]=\left[0 \\ 0 \\ 0\right]    \end{gathered}$$

$2x_1+2x_2+x_3=0,  x_1+x_2+2x_3=0$

$x_1+x_2=0,x_3=0$

$$\begin{gathered} E_0=span\left(\left[1 \\ -1 \\ 0\right]\right) \end{gathered}$$

$ \mathrm{λ}=1$solved,

$\left(A-I\right)x=0$

$$\begin{gathered} \left[1            2            1 \\ 0           -1            0 \\ 1             1              1\right]  \left[x_1 \\  x_2 \\ {x}_3\right]=\left[0 \\ 0 \\ 0\right] \end{gathered}$$

$$\begin{gathered} x_1+2x_2+x_3=0,  x_1+x_2+x_3=0 \\ {x}_1+x_3=0,x_2=0 \end{gathered}$$

$$\begin{gathered} E_0=span\left(\left[1 \\ 0 \\ -1\right]\right) \end{gathered}$$

Finally,

$ \mathrm{λ}=3$solved,

$$\begin{gathered} \left[-1            2                 1 \\ 0           -3            0 \\ 1             1              -1\right]  \left[x_1 \\  x_2 \\ {x}_3\right]=\left[0 \\ 0 \\ 0\right] \end{gathered}$$

$\left(A-3I\right)x=0$

$$\begin{gathered} -x_1+2x_2+x_3=0,  x_1+x_2-x_3=0 \\ {x}_1-x_3=0,x_2=0 \end{gathered}$$

$$\begin{gathered} E_0=span\left(\left[1 \\ 0 \\ 1\right]\right) \end{gathered}$$

$$\begin{gathered} x_0=\left[\begin{array}{c}0\\ 2\\ 0\end{array}\right]=-2\left[1 \\ -1 \\ 0\right]+\frac{1}{2}\left[1 \\ 0 \\ -1\right]+\frac{1}{2}\left[\begin{array}{c}1\\ 0\\ 1\end{array}\right] \end{gathered}$$

$$\begin{gathered} A^t{x}_0=-2\left[1 \\ -1 \\ 0\right]+\frac{1}{2}.2^t\left[1 \\ 0 \\ -1\right]+\frac{1}{2}.3^t\left[1 \\ 0 \\ 1\right]=\frac{1}{2}\left[-4+2^t +3^t  \\           4 \\     -2^t+3^t\right] \end{gathered}$$

Hence,

$$\begin{gathered} A^t{x}_0=-2\left[1 \\ -1 \\ 0\right]+\frac{1}{2}.2^t\left[1 \\ 0 \\ -1\right]+\frac{1}{2}.3^t\left[1 \\ 0 \\ 1\right]=\frac{1}{2}\left[-4+2^t +3^t  \\           4 \\     -2^t+3^t\right] \end{gathered}$$