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Multiplicity of Eigenvalues

  1. Apr 17, 2012 #1
    1. The problem statement, all variables and given/known data
    The matrix [itex]A = \begin {bmatrix} 0 & -1 & 0 \\ 0 & -1 & 0 \\ 0 & 1 & 0 \end{bmatrix}[/itex] has two real eigenvalues, one of multiplicity 2 and one of multiplicity 1. Find the eigenvalues and a basis of each eigenspace.


    2. Relevant equations
    N/A


    3. The attempt at a solution
    I've done cofactor expansion to come up with the equation [itex] - \lambda^3 - \lambda^2[/itex], and that the eigenvalues are therefore -1 and 0, but I don't know how to determine the multiplicity of each. I've looked it up and have gotten nowhere. I can do the rest of the problem, I just don't know how to get these multiplicities.

    EDIT: Working on the problem further, and I got the basis for -1. There are supposed to be two vectors for 0, however, but the RREF of the matrix is [itex]\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}[/itex]. How does that translate to two eigenvectors?

    Thanks in advance for any help!
     
    Last edited: Apr 17, 2012
  2. jcsd
  3. Apr 17, 2012 #2

    Mark44

    Staff: Mentor

    That's not an equation. The equation is [itex] - \lambda^3 - \lambda^2[/itex] = 0, or ##\lambda^2(\lambda + 1) = 0##
    Can you see that one of the eigenvalues has multiplicity 2 and the other has multiplicity 1?
     
  4. Apr 18, 2012 #3
    Ah, yes, I just couldn't simplify that properly; sorry! Thank you!

    [STRIKE]But now the second part -- how does that matrix equate to two eigenvectors?[/STRIKE]

    I'm an idiot; I figured it out. Thanks again!
     
    Last edited: Apr 18, 2012
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