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Homework Help: Confused on this defective matrix problem

  1. Apr 24, 2010 #1
    Make matrix defective if possible and identify the values of alpha.

    [tex]\begin{bmatrix}
    3\alpha & 1 & 0\\
    0 & \alpha & 0\\
    0 & 0 & \alpha
    \end{bmatrix}[/tex]

    Skipping the boring stuff we obtain [itex](\alpha-\lambda)^2(3\alpha-\lambda)=0[/itex] as the characteristic polynomial.

    [tex]\lambda_1=\lambda_2=\alpha[/tex] and [tex]\lambda_3=3\alpha[/tex]
    For lambda being alpha
    [tex]\begin{bmatrix}
    2\alpha & 1 & 0\\
    0 & 0 & 0\\
    0 & 0 & 0
    \end{bmatrix}\Rightarrow \begin{bmatrix}
    1 & \frac{-1}{2\alpha} & 0\\
    0 & 0 & 0\\
    0 & 0 & 0
    \end{bmatrix}\Rightarrow x_2\begin{bmatrix}
    \frac{-1}{2\alpha}\\
    1\\
    0
    \end{bmatrix}+x_3\begin{bmatrix}
    0\\
    0\\
    1
    \end{bmatrix}[/tex]

    For this lambda value, the matrix can't be defective? Not sure though.

    For lambda being 3alpha
    [tex]\begin{bmatrix}
    0 & 1 & 0\\
    0 & -2\alpha & 0\\
    0 & 0 & -2\alpha
    \end{bmatrix}\Rightarrow \begin{bmatrix}
    0 & 1 & 0\\
    0 & 0 & 1\\
    0 & 0 & 0
    \end{bmatrix}\Rightarrow x_1\begin{bmatrix}
    1\\
    0\\
    0
    \end{bmatrix}[/tex]

    And for this one, alpha can be any value and the matrix will be defective.
     
    Last edited: Apr 25, 2010
  2. jcsd
  3. Apr 25, 2010 #2

    gabbagabbahey

    User Avatar
    Homework Helper
    Gold Member

    A 3x3 matrix is defective it has fewer that 3 distinct, linearly independent eigenvectors. How many distinct, linearly independent eigenvectors does this matrix have?
     
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