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Generalized eigenvector?
The problem I'm doing asks me to solve the normal system x'=Ax, but A is an unknown 2x2 matrix. However, I'm given some initial values and:
$\displaystyle
A \left(\begin{array}{r}1\\4\end{array}\right) = 4 \left(\begin{array}{r}1\\4\end{array}\right)
$
OK, I know this says 4 is an eigenvalue with corresponding eigenvector (1,4)
$\displaystyle
A \left(\begin{array}{r}0\\1\end{array} \right) = 4 \left(\begin{array}{r}0\\1\\\end{array} \right)
+ \left( \begin{array}{r}1\\4\end{array}\right)
$
But I'm not sure what to make of this.
My current approach is to find $\displaystyle e^{At} $ so I need a second, maybe generalized? eigenvector and corresponding eigenvalue.
edit: Never mind, I figured I can just solve for A