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Eigenspaces homework problem

  1. Jul 29, 2012 #1
    1. The problem statement, all variables and given/known data
    Find the Eigenspace of the following matrix:
    [tex]\begin{bmatrix}
    1 & 3 \\
    4 & -3
    \end {bmatrix}[/tex]
    I'm skipping a few steps but the Eigenvalues are -5 and 3. Let's starts with -5. Skip a few more steps, I know I'm right, just trust me.
    We now have the following matrix:
    [tex]\begin{bmatrix}
    -6 & -3 \\
    -4 & -2
    \end {bmatrix}[/tex]
    Then you find the null space, which starts with putting it in reduced row echelon form:
    [tex]\begin{bmatrix}
    -6 & -3 \\
    0 & 0
    \end {bmatrix}[/tex]
    you can reduce that further to
    [tex]\begin{bmatrix}
    -2 & -1 \\
    0 & 0
    \end {bmatrix}[/tex]
    This is where I'm confused. This nullspace calculator http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi
    says that the basis of the null space is
    [tex]\begin{bmatrix}
    -1 \\
    2
    \end {bmatrix}[/tex]
    My textbook confirms that. How do I get from here
    [tex]\begin{bmatrix}
    -2 & -1 \\
    0 & 0
    \end {bmatrix}[/tex]
    to there
    [tex]\begin{bmatrix}
    -1 \\
    2
    \end {bmatrix}[/tex]
    I would think you would just eliminate the 2nd row and transpose the first row but that would give.
    [tex]\begin{bmatrix}
    -2 \\
    -1
    \end {bmatrix}[/tex]
     
    Last edited: Jul 29, 2012
  2. jcsd
  3. Jul 29, 2012 #2

    LCKurtz

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    Re: Eigenspaces

    The matrix corresponds to -2x - y = 0 or y = -2x So$$
    \begin{bmatrix}x\\ y \end{bmatrix}=\begin{bmatrix}x\\ -2x \end{bmatrix}
    =x\begin{bmatrix}1\\ -2 \end{bmatrix} $$
    Any nonzero constant works, so take ##x=-1## to get their answer.
     
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