Question

Consider a dynamical systemwith two components. The accompanying sketch shows the initial state vector${\overrightarrow{\mathbf{x}}}_{\mathbf{0}}$and two eigen vectors$\overrightarrow{{\mathbf{\upsilon }}_{\mathbf{1}}}\mathbf{ }\mathbf{ }\mathit{a}\mathit{n}\mathit{d}\mathbf{ }\mathbf{ }\overrightarrow{{\mathbf{\upsilon }}_{\mathbf{2}}}$of A (with eigen values $\overrightarrow{{\mathbf{\lambda }}_{\mathbf{1}}}\mathbf{}\mathbf{\text{and}}\mathbf{}\overrightarrow{{\mathbf{\lambda }}_{\mathbf{2}}}$respectively). For the given values of$\overrightarrow{{\mathbf{\lambda }}_{\mathbf{1}}}\mathbf{}\mathbf{\text{and}}\mathbf{}\overrightarrow{{\mathbf{\lambda }}_{\mathbf{2}}}$, draw a rough trajectory. Consider the future and the past of the system.

$\overrightarrow{{\mathbf{\lambda }}_{\mathbf{1}}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{9}\mathbf{,}\overrightarrow{{\mathbf{\lambda }}_{\mathbf{2}}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{9}$

Short answer

So, the required solution is$A^t{x}_0=0.9^t{x}_0$_.

Step-by-step solution

Define the eigenvector

Eigenvector:An eigenvector of $\mathbf{A}$is a nonzero vector $\mathbf{v}$in${\mathbf{R}}_{\mathbf{n}}$such that$\mathbf{Av}\mathbf{=}\mathbf{\lambda v}\mathbf{}$, for some scalar$\mathbf{⅄}$.

Note the given data

It is given that:

$$\begin{gathered} \overrightarrow{{\lambda }_1}=0.9, \\ \overrightarrow{{\lambda }_2}=0.9 \end{gathered}$$

Given graph is:

Finding the required matrix

We have:

$$\begin{gathered} A{\upsilon }_1=0.9{\upsilon }_1 \\ A{\upsilon }_2=0.9{\upsilon }_2 \end{gathered}$$

For$x_0=\alpha {\upsilon }_1+\beta {\upsilon }_2$,We have:

$$\begin{gathered} A{x}_0=A(\alpha {\upsilon }_1+\beta {\upsilon }_2) \\ =\alpha A{\upsilon }_1+\beta A{\upsilon }_2 \\ =0.9\alpha {\upsilon }_1+0.9\beta {\upsilon }_2 \\ =0.9x_0 \end{gathered}$$

Therefore,$A^t{x}_0=0.9^t{x}_0$.

Hence, the solutions is$A^t{x}_0=0.9^t{x}_0$._.