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Jordan Canonical Form

  1. Oct 16, 2012 #1
    1. The problem statement, all variables and given/known data
    Let A be an nxn matrix (of real or complex components) and

    [tex]J=\left(\begin{array}{c} λ & 1 & 0 & 0\\
    0 & λ & 1 & 0\\
    & & ... & \\
    0 & 0 & λ & 1\\
    0 & 0 & 0 & λ
    &\end{array}\right) \,with\, λ \in ℂ[/tex]


    Show that there is
    [tex]S = \left(\begin{array}{c} v1 & v2 & ... & v_n &\end{array}\right)
    \,with\, v1, v2, ...,v_n \inℂ [/tex]
    such that A = SJS-1 if and only if:
    (A - λI)v1 = 0
    (A - λI)vi+1 = vi , for i=1,2,....,n-1


    2. Relevant equations



    3. The attempt at a solution
    My first step was A = SJS-1 ⇔ AS = SJ
    Now, developing the right side I get SJ = [λv1 , v1 + λv2 ... vn-1 + λvn ]

    So, column by column I get: Av1 = λv1 ⇔ (A-λ)v1 = 0
    Av2 = v1 + λv2 ⇔ (A-λ)v2 = v1

    and extending, I get
    (A - λI)vi+1 = vi , for i=1,2,....,n-1

    My only question is, does this solve the problem? I thought that to prove a ⇔ b , I had to prove a [itex]\Rightarrow[/itex] b and b [itex]\Rightarrow[/itex] a, but it seems to me that this proves both ways.
     
  2. jcsd
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