Prof. Alejandro Ucan-Puc
Consider the system $$X'= \left(\begin{array}{cc} 1 & 2 \\ -2 & -3 \end{array}\right) X.$$ We want to build a general solution. The characteristic polynomial is: $$\lambda^2-2\lambda +1=0,$$ so the only eigenvalue is $\lambda=1.$ Its eigenvector is $$(A+I)=\left(\begin{array}{cc} 2 & 2 \\ -2 & -2 \end{array}\right)\Rightarrow 2v_1+2v_2=0\Rightarrow v_2=-v_1.$$
In this case, we say that the eigenvector is not enough to build a general solution. We need to find a second vector. In order to do this, we need to find $w$ such that $$(A-\lambda I)w=v.$$
Solve the system $$X'= \left(\begin{array}{cc} 1 & 2 \\ -2 & -3 \end{array}\right) X.$$
We have that $$(A+I)w=v\Rightarrow \left(\begin{array}{cc} 2 & 2 \\ -2 & -2 \end{array}\right) \left(\begin{array}{c} w_1 \\ w_2 \end{array}\right) =\left(\begin{array}{c} 1 \\ -1 \end{array}\right),$$ which reduces to the equation $$2w_1+2w_2=1\Leftrightarrow 2w_1=1+2w_2.$$ We can take the value $w_2=0,$ and my other vector would be $$\left(\begin{array}{c} 1/2 \\ 0 \end{array}\right)$$ Thus my general solution is $$X=c_1e^{-t}\left(\begin{array}{c} -1 \\ 1 \end{array}\right)+c_2te^{-t}\left(\begin{array}{c} 1/2 \\ 0 \end{array}\right).$$
Find the general solution for $$X'=\left(\begin{array}{cc} 3 & -18 \\ 2 & -9 \end{array}\right)X.$$
Consider the system $$X'= \left(\begin{array}{cc} 6 & -1 \\ 5 & 4 \end{array}\right) X.$$
Its characteristic polynomial is $$\lambda^2-10\lambda+29=0\Rightarrow \lambda_1=5+2i\quad \lambda_2=5-2i.$$
Its eigenvectors are $$v_1=\left(\begin{array}{c} 1 \\ 1-2i \end{array}\right)\quad v_2=\left(\begin{array}{c} 1 \\ 1+2i \end{array}\right)$$
As we want real functions, we need to rewrite the eigenvectors in order to get real functions.
Let $$w_1=\left(\begin{array}{c} 1 \\ 1 \end{array}\right) \quad w_2=\left(\begin{array}{c} 0 \\ -2 \end{array}\right)$$ Thus my general solution is $$X=c_1e^{(5+2i)t}\left(\begin{array}{c} 1 \\ 1-2i \end{array}\right)+c_2e^{(5-2i)t}\left(\begin{array}{c} 1 \\ 1+2i \end{array}\right).$$
Find the general solution for $$X'=\left(\begin{array}{cc} -1 & 2 \\ -1 & 1 \end{array}\right)X.$$
Defective matrix systems, chains of generalized eigenvectors (A - lambda I)v2 = v1, complex conjugate eigenvalue solutions, and phase plane portraits.
Next Session: The Laplace Transform