Question

Suppose 7' = AT, where A is the 2 x 2 matrix below. A= (1 1 1 3

Answer 1

(a) We have the matrix A as

$\small A=\begin{pmatrix} 1 & -1\\ 1 &3 \end{pmatrix}$

The eigenvalues of A are simply the roots of its characteristic equation, which is

$\small \det(A-xI)=\begin{vmatrix} 1-x & -1\\ 1 &3-x \end{vmatrix}=(1-x)(3-x)+1=x^2-4x+4=(x-2)^2$

Hence the roots of the characteristic equation are $\small x=2,2$

Thus, the matrix has a single eigenvalue, 2, with multiplicity 2.

Now, we have to find the eigenvectors corresponding to this eigenvalue, which are given by

$\small Av=2v\Rightarrow \begin{pmatrix} 1 & -1\\ 1 &3 \end{pmatrix}\begin{pmatrix} v_1\\ v_2 \end{pmatrix}=\begin{pmatrix} 2v_1\\ 2v_2 \end{pmatrix}\Rightarrow \begin{pmatrix} v_1-v_2 \\ v_1+3v_2 \end{pmatrix}=\begin{pmatrix} 2v_1\\ 2v_2 \end{pmatrix}$

Now, equating the components together, we get the same equation :-

$\small \Rightarrow \begin{matrix} v_1-v_2=2v_1\\ v_1+3v_2=2v_2 \end{matrix}\Rightarrow v_1+v_2=0 \Rightarrow v_1=-v_2$

Thus, the general form of the eigenvector is given by

$\small \begin{pmatrix} v_1\\ v_2 \end{pmatrix}=\begin{pmatrix} -v_2\\ v_2 \end{pmatrix}=v_1\begin{pmatrix} -1\\ 1 \end{pmatrix}$

this completes part (a)

(b)

For repeated eigenvalues, the general solution of the system is given by

$\small x=c_1e^{\lambda\:t}\eta+c_2e^{\lambda\:t}\left(t\eta+\rho \right)$

where

$\small \lambda =2$

$\small \eta = \begin{pmatrix} -1\\ 1 \end{pmatrix}$

and $\small \rho$ is a solution of the equation

$\small \left(A-\lambda\:I\right)\rho=\eta$

So,

$\small \left(A-2I\right)\rho=\begin{pmatrix} -1\\ 1 \end{pmatrix}$

Let rho = (p1,p2) so we have

$\small \begin{pmatrix} -1 &-1 \\ 1& 1 \end{pmatrix}\begin{pmatrix} \rho_1\\ \rho_2 \end{pmatrix}=\begin{pmatrix} -1\\ 1 \end{pmatrix}\Rightarrow \begin{matrix} -\rho_1-\rho_2=-1\\ \rho_1+\rho_2=1 \end{matrix}\Rightarrow \rho_1+\rho_2=1$

So, one possible solution is obtained by putting p2 = 0, so

$\small \rho=\begin{pmatrix} 1\\ 0 \end{pmatrix}$

Thus, we have the general solution of the system as

$\small x=c_1e^{\lambda\:t}\eta+c_2e^{\lambda\:t}\left(t\eta+\rho \right)$

$\small \Rightarrow x=c_1e^{2t}\begin{pmatrix} -1\\ 1 \end{pmatrix}+c_2e^{2t}\left(t\begin{pmatrix} -1\\ 1 \end{pmatrix}+\begin{pmatrix} 1\\ 0\end{pmatrix} \right)$

Simplifying we get

$\small \Rightarrow x=c_1e^{2t}\begin{pmatrix} -1\\ 1 \end{pmatrix}+c_2e^{2t}\begin{pmatrix} 1-t\\ t \end{pmatrix}$

which is our required solution.

(c) and the phase portrait of the system is

3.0 تی 20 1.0 0.0 ا -1.0 - 20 -3.0 - -3 - -1 1 اليا

where y is along the vertical axis and x is along the horizontal axis