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Math Help - Eigenvalues part 2

  1. #1
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    Eigenvalues part 2

    i need to find the eigenvalues and vectors of this matrix
    [0 1]
    [-1 0]
    lamda - A gives me this:
    [Lamda -1]
    [1 Lamda]

    Solve the det of this to get lamda squared +1

    Only solution is imaginary, i and -i
    The matrix of ilamda - A:
    [i -1]
    [1 i]

    Not sure what to do here. Never worked with imaginary numbers in a matrix. Can someone show me how to solve this. Thanks
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  2. #2
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    Quote Originally Posted by PensFan10 View Post
    i need to find the eigenvalues and vectors of this matrix
    [0 1]
    [-1 0]
    lamda - A gives me this:
    [Lamda -1]
    [1 Lamda]

    Solve the det of this to get lamda squared +1

    Only solution is imaginary, i and -i
    The matrix of ilamda - A:
    [i -1]
    [1 i]

    Not sure what to do here. Never worked with imaginary numbers in a matrix. Can someone show me how to solve this. Thanks
    Is your matrix over \mathbb{C}? Or over \mathbb{R}? If the latter, then there are no eigenvectors. If the first, then it is the same process as working with real numbers.
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  3. #3
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    its over c. how do you rref with i?
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  4. #4
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    You "rref" with complex number exactly like with real numbers! Just remember to multiply and divide correctly!

    But I would bother with "rref" here. Just use the definition of "eigenvalue": If \lambda is an eigenvalue of A, then there exist a non-zero vector, v, such that Av= \lambda v and the eigenvectors are the vectors satisfying that.

    Since i is an eigenvalue, you must have \begin{bmatrix} 0 & 1 \\ -1 & 0\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}= i\begin{bmatrix} x \\ y \end{bmatrix}

    \begin{bmatrix}y \\-x\end{bmatrix}= \begin{bmatrix}ix \\ iy\end{bmatrix}.

    That gives the two equations y= ix and -x= iy which, since 1/i= -i, are really the same equation. From y= ix, we can write \begin{bmatrix}x \\ y\end{bmatrix}= \begin{bmatrix}x \\ ix\end{bmatrix}= x\begin{bmatrix}1 \\ i\end{bmatrix}.

    The eigenvectors corresponding to eigenvalue i are multiples of \begin{bmatrix}1 \\ i\end{bmatrix}.

    Now you find the eigenvectors corresponding to eigenvalue -i.
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