Question

consider the dynamical system

$\overrightarrow{\mathbf{x}}\mathbf{(}\mathit{t}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{=}\left[\begin{array}{cc}1.1& 0\\ 0& \lambda \end{array}\right]\overrightarrow{\mathbf{x}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$

Sketch a phase portrait of this system for the given values of:

Short answer

The sketch a phase portrait of this system for the given values of $\lambda$is 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:

$$\begin{gathered} A^t{x}_0=\left[\begin{array}{cc}{\left(1,1\right)}^t& 0\\ 0& 1\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right] \\ =\left[\begin{array}{c}{\left(1.1\right)}^t{x}_1\\ x_2\end{array}\right] \end{gathered}$$

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