Hello, I'm trying to find the eigenvector(s) of the following matrix:
![\[ \left( \begin{array}{cc} 2 & -4 \\ 0 & 2 \end{array} \right) \]](/CBB/latexrender/pictures/80389a4757e50f300b7f5adea08d51d3.png "\[ \left( \begin{array}{cc} 2 & -4 \\ 0 & 2 \end{array} \right) \]")
I already know that it's a doubled eigenvalue of
, obvious given that it's a diagonal matrix. However, I'm having trouble finding the eigenvectors, because when solving:
![\[ \left( \begin{array}{cc} 2-\lambda & -4 \\ 0 & 2-\lambda \end{array} \right) \rightarrow \left( \begin{array}{cc} 0 & 4 \\ 0 & 0 \end{array} \right) \]](/CBB/latexrender/pictures/0c0fca50345aa4eab7b8f732ad2601c5.png "\[ \left( \begin{array}{cc} 2-\lambda & -4 \\ 0 & 2-\lambda \end{array} \right) \rightarrow \left( \begin{array}{cc} 0 & 4 \\ 0 & 0 \end{array} \right) \]")
You can't really do the system of equations because you just end up with:
![\[ \left( \begin{array}{cc} 0 & 4 \\ 0 & 0 \end{array} \right) \cdot \left( \begin{array}{c} \eta_{1} \\ \eta_{2} \end{array} \right) = \left( \begin{array}{cc} 0 \\ 0 \end{array} \right)\]](/CBB/latexrender/pictures/edf2d9794fc6e3434f2078076fe7f0e5.png "\[ \left( \begin{array}{cc} 0 & 4 \\ 0 & 0 \end{array} \right) \cdot \left( \begin{array}{c} \eta_{1} \\ \eta_{2} \end{array} \right) = \left( \begin{array}{cc} 0 \\ 0 \end{array} \right)\]")
.
Which just gives you
which isn't really anything you can use to solve for the eigenvector
.
Could anyone tell me what the eigenvectors are, and more importantly, how they were found?
This is in the context of finding the general solution to the system
![\[ X' = \left( \begin{array}{cc} 2 & -4 \\ 0 & 2 \end{array} \right) X \]](/CBB/latexrender/pictures/dbc6a4a90fe89e68ea49832ba7a61b58.png "\[ X' = \left( \begin{array}{cc} 2 & -4 \\ 0 & 2 \end{array} \right) X \]")
though I already know the answer is
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