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Repeated eigenvalues

  1. Apr 17, 2012 #1
    Hi there!

    Let A be a square matrix of order n.
    It is well known that if we have n distinct eigenvalues then we surely have n distinct eigenvectors. But if there are repeated eigenvalues then the tow possibilities may happen.
    My question is: How can I know that do the eigenvectors are distinct or not?

    Thank you very much.
     
  2. jcsd
  3. Apr 17, 2012 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    The only way to be certain is by trying to find the eigenvectors!

    For example both
    [tex]\begin{bmatrix}a & 0 \\ 0 & a\end{bmatrix}[/tex]
    and
    [tex]\begin{bmatrix} a & 1 \\ 0 & a\end{bmatrix}[/tex]
    have a as a double eigenvalue.
    To find the eigenvectors, we need to solve
    [tex]\begin{bmatrix}a & 0 \\ 0 & a\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}= \begin{bmatrix}ax \\ ay\end{bmatrix}[/tex]
    and
    [tex]\begin{bmatrix}a & 1 \\ 0 & a\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix}= \begin{bmatrix}ax \\ ay\end{bmatrix}[/tex]

    The first gives the two equations ax= ax and ay= ay. Clearly, those are true for all x and y- any vector in R2 and, in particular <1, 0> and <0 1>, which are independent, are eigenvectors.

    The second gives the two equations ax+ y= ax and ay= ay. The second equation is true for any y but the first equation reduces to y= 0. Given that a can be anything but we have that all eigenvectors are of the form <a, 0>, a one dimensional space so we have only one "independent" eigenvector.
     
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