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Math Help - Eigenvector with complex eigenvalues

  1. #1
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    Eigenvector with complex eigenvalues

    I have to find the eigenvalues and eigenvectors for this matrix:

    \begin{pmatrix}4&2i\\2i&6\end{pmatrix}

    I found eigenvalues of \lambda=5\pm\sqrt{3}i

    I'm having trouble finding the eigenvectors

    ker\begin{pmatrix}-1\pm\sqrt{3}i&2i\\2i&1\pm\sqrt{3}i\end{pmatrix}

    I know how to find eigenvectors, but this one seems confusing because of the complex eigenvalues
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by acevipa View Post
    I have to find the eigenvalues and eigenvectors for this matrix:

    \begin{pmatrix}4&2i\\2i&6\end{pmatrix}

    I found eigenvalues of \lambda=5\pm\sqrt{3}i

    I'm having trouble finding the eigenvectors

    ker\begin{pmatrix}-1\pm\sqrt{3}i&2i\\2i&1\pm\sqrt{3}i\end{pmatrix}

    I know how to find eigenvectors, but this one seems confusing because of the complex eigenvalues
    Just do what's natural suppose that \begin{bmatrix}z_1\\ z_2\end{bmatrix}\in\mathbb{C}^2 is an eigenvector, then: \begin{bmatrix}4 & 2i\\ 2i &6\end{bmatrix}\begin{bmatrix}z_1\\ z_2\end{bmatrix}=\begin{bmatrix}(5+\sqrt{3}i)z_1\\ (5+\sqrt{3}i)z_2\end{bmatrix}. Expand this and solve.
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