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Homework Help: Eigenspace help

  1. Apr 14, 2010 #1
    [tex]\begin{bmatrix}
    3 & 2\\
    4 & 1
    \end{bmatrix}[/tex]
    [tex]det(A-\lambda I)=\begin{vmatrix}
    3-\lambda & 2\\
    4 & 1-\lambda
    \end{vmatrix}=(3-\lambda)(1-\lambda)-8=\lambda^2-4\lambda-5[/tex]
    [tex]\lambda_{1}=5[/tex] and [tex]\lambda_{2}=-1[/tex]
    When [tex]\lambda=5[/tex], [tex]\begin{bmatrix}
    -2 & 2\\
    4 & -4
    \end{bmatrix}\Rightarrow \begin{bmatrix}
    1 & -1\\
    0 & 0
    \end{bmatrix}[/tex]
    The eigenspace for [tex]\lambda_{1}[/tex] is [tex]\begin{bmatrix}
    1\\
    1
    \end{bmatrix}[/tex]
    When [tex]\lambda=-1[/tex], [tex]\begin{bmatrix}
    4 & 2\\
    4 & 2
    \end{bmatrix}\Rightarrow \begin{bmatrix}
    1 & 0\\
    0 & 1
    \end{bmatrix}[/tex]
    The eigenspace for[tex]\lambda_{2}[/tex] is [tex]\begin{bmatrix}
    0\\
    0
    \end{bmatrix}[/tex]

    I don't know what is going wrong but my second Eigenspace is wrong compared to the books answer which is [tex]\begin{bmatrix}
    1\\
    -2
    \end{bmatrix}[/tex]
     
  2. jcsd
  3. Apr 14, 2010 #2

    Mark44

    Staff: Mentor

    Re: Eigenspace

    Your mistake is above. The [4 2; 4 2] matrix doesn't row reduce to the identity matrix. Try again.
    Each matrix for calculating the eigenspace can't reduce to the identity; otherwise its determinant would not be zero.
     
  4. Apr 14, 2010 #3
    Re: Eigenspace

    I had a -2 entered into my calc.
     
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