Generalized Eigenvectors for repeated eigenvalues

class: center, middle, inverse, title-slide

.title[
# Generalized Eigenvectors for repeated eigenvalues
]
.subtitle[
## Session 08
]
.author[
### Alejandro Ucan
]
.date[
### 2023-10-29
]

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# Goals

* Understand the concept of generalized eigenvectors. <br/><br/>
* Understand the solution in the case of complex eigenvalues. <br/><br/>
* Understand the phase plane. <br/><br/>

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# Generalized eigenvectors

> 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 `$$(A+I)=\left(\begin{array}{cc} 2 & 2 \\ -2 & -2 \end{array}\right)\Rightarrow 2v_1+2v_2=0\Rightarrow v_2=-v_1.$$` <br/>
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 `$$(A-\lambda I)w=v.$$`
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## Continuation example

> Solve the system `$$X'= \left(\begin{array}{cc} 1 & 2 \\ -2 & -3 \end{array}\right) X.$$` <br/><br/> 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 `$$X=c_1e^{-t}\left(\begin{array}{c} -1 \\ 1 \end{array}\right)+c_2te^{-t}\left(\begin{array}{c} 1/2 \\ 0 \end{array}\right).$$`

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### Example 2:

> Find the general solution for `$$X'=\left(\begin{array}{cc} 3 & -18 \\ 2 & -9 \end{array}\right)X.$$`

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# Complex Eigenvalues

> Consider the system `$$X'= \left(\begin{array}{cc} 6 & -1 \\ 5 & 4 \end{array}\right) X.$$` <br/><br/> Its characteristic polynomial is `$$\lambda^2-10\lambda+29=0\Rightarrow \lambda_1=5+2i\quad \lambda_2=5-2i.$$` <br/><br/> Its eigenvectors are `$$v_1=\left(\begin{array}{c} 1 \\ 1-2i \end{array}\right)\quad v_2=\left(\begin{array}{c} 1 \\ 1+2i \end{array}\right)$$`
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## Rewriting the eigenvectors

> As we want real functions, we need to rewrite the eigenvectors in order to get real functions. <br/><br/> 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 `$$X=c_1e^{(5+2i)t}\left(\begin{array}{c} 1 \\ 1-2i \end{array}\right)+c_2e^{(5-2i)t}\left(\begin{array}{c} 1 \\ 1+2i \end{array}\right).$$`
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### Example 3:

> Find the general solution for `$$X'=\left(\begin{array}{cc} -1 & 2 \\ -1 & 1 \end{array}\right)X.$$`

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# Phase Plane