I have problem and need your help. I must draw phase portraits of dynamical system which looks like this: $$\dot{x}_{1}(t) = -x_{1}(t) + x_{2}(t)$$ $$\dot{x}_{2}(t) = -x_{2}(t)$$ I know that the first I sould get eigenvector and eigenvalue of matrix but I'm not soure. Is this matrix will looks like this:
$$\begin{matrix} -1 & 1 \\ -1 & 0 \\ \end{matrix}$$ Am I right? What sould I do next?
On
First thing should be to check if it's Hamiltonian which makes the whole process much easier as you just sketch level sets.
Find the stationary points, then classify them using the trace and determinant of the linearized system at those points. You can find the eigenvectors to find the orientation of sources/sinks/saddles/centres if it's not obvious from plugging in some values.
This should get you quite some way to a complete portrait.
Not sure I understand what you are asking exactly but, if you want to draw a phase portrait, draw a phase portrait:
The general approach is to delineate the regions of the $(x_1,x_2)$-plane where $x'_1$ and $x'_2$ have a constant sign, and to deduce the variations of the solutions from this decomposition. These regions are limited by the so-called nullclines, which are the lines where $x'_1=0$ or $x'_2=0$.
In the present case, $x'_1=0$ corresponds to the first diagonal $x_2=x_1$ and $x'_2=0$ corresponds to the horizontal axis $x_2=0$. For example, at every point in the region $0<x_2<x_1$ (North-East to East angular sector), the dynamics points to the South-East ($x'_1<0$, $x'_2<0$). Likewise for the three other angular sectors and the four halflines which delimit them (whose union is the union of the nullclines), hence the diagram above.
Finally, in case the system you are interested in is actually $$x'_1=-x_1+x_2\qquad x'_2=-x_1$$ the same procedure applies, and yields the phase portrait below.