Question

Consider the dynamical system $\overrightarrow{\mathbf{x}}\left(t+1\right)\mathbf{=}\left[\begin{array}{cc}1.1& 0\\ 0& ⅄\end{array}\right]\overrightarrow{\mathbf{X}}\left(t\right)$.

Sketch a phase portrait of this system for the given values of $\mathit{\lambda }$:

$\mathit{\lambda }\mathbf{=}\mathbf{1}$

Short answer

The sketch a phase portrait of this system for the given values ofis shown as below:

Step-by-step solution

Definition of matrix

A diagonal matrix is a matrix in which the entries outside the main diagonal are all zero.

Sketch the phase portrait

Consider the given matrix,

$A=\left[\begin{array}{cc}1.1& 0\\ 0& 1\end{array}\right]$

Clearly, A is diagonal, so for any:

$X_{0=}\left[\begin{array}{c}{X}_1\\ X_2\end{array}\right]$

, we have:

$$\begin{gathered} A^t{X}_0=\left[\begin{array}{cc}{(1.1)}^t& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}{X}_1\\ X_2\end{array}\right] \\ =\left[\begin{array}{cc}{(1.1)}^t& X_1\\ & X_2\end{array}\right] \end{gathered}$$

So, the required sketch is shown as below using the technology.