Jordan Basis for Differential Operator

  1. 1. The problem statement, all variables and given/known data
    Let [itex] V = P_n(\textbf{F}) [/itex]. Prove the differential operator D is nilpotent and find a Jordan basis.

    2. Relevant equations
    [itex] D(Ʃ a_k x^k ) = Ʃ k* a_k * x^{k-1} [/itex]

    3. The attempt at a solution
    I already did the proof of D being nilpotent, which was easy. But we haven't covered what a "Jordan basis" is in class and it's not in either of my textbooks. I know what Jordan Canonical Form is, and Jordan blocks, but I don't know what a Jordan basis is.

    Earlier I did a problem that showed that the matrix form of the differential operator on polynomials of order 2 or less. It was
    [itex]
    \left[
    \begin{array}{ c c }
    0 & 1 & 0 \\
    0 & 0 & 2 \\
    0 & 0 & 0
    \end{array} \right]
    [/itex]
    Is that the kind of basis they're looking for here?
     
  2. jcsd
  3. morphism

    morphism 2,020
    Science Advisor
    Homework Helper

    No - in a Jordan basis, all entries in the superdiagonal (i.e. the line above the diagonal) have to be either 1 or zero.
     
  4. So do I need something like

    \begin{array}{ccc}
    0 & 1 & 0 & \dots & 0 \\
    0 & 0 & 1 & \dots & 0 \\
    \dots \\
    0 & 0 & 0 & \dots & 1 \\
    0 & 0 & 0 & \dots & 0 \end{array}

    as an n-vector Jordan basis for the polynomials of order up to n?
     
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