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Proof of an upper triangular matrix - theory

  1. Jan 24, 2012 #1
    1. The problem statement, all variables and given/known data
    Suppose that A is a 3x3 matrix, having characteristic polynomial [itex](λ-x)^{3}[/itex]. Suppose that there is a vector [itex]v[/itex] in [itex]F^3[/itex] which is not in [itex]Ker(A-λI)^2[/itex]. You may assume that [itex](A-λI)^3=0[/itex]. Show that [itex](A-λI)^2v, (A-λI)v, v[/itex] are linearly independent. If Y is the matrix having columns [itex](A-λI)^2v, (A-λI)v, v[/itex] show that if λ=3 then

    [tex]Y^{-1}AY = \begin{pmatrix}3 & 1 & 0\\0 & 3 & 1\\0 & 0 & 3\end{pmatrix}[/tex]


    3. The attempt at a solution
    I have already proved that [itex](A-λI)^2v, (A-λI)v, v[/itex] are linearly independent by using the fact that [itex](A-λI)^3=0[/itex] and [itex](A-λI)^2≠0[/itex]. I have no idea where to start in proving the matrix of [itex]Y^{-1}AY[/itex].

    The hints our prof gave us was that we want:
    [itex]Av_1=3(A-3I)^2v = A(A-3I)^2v[/itex]

    [itex](A-3I)^3v=0 ~so~ (A-3I)(A-3I)^2v=0~~=>~~(A-3I)v_1=0[/itex]

    For [itex]v_2=(A-3I)v ~~~~ (A-3I)v_2=v_1[/itex]

    But I'm not quite sure how this helps. Thanks!
     
    Last edited by a moderator: Jan 24, 2012
  2. jcsd
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