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Math Help - Finding minimal polynomial

  1. #1
    Junior Member
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    Finding minimal polynomial

    Hi everyone,

    could you please help me with the following?

    Let  n \geq 1 and let  V_{n} be the subspace of  \mathbb{R} [x,y] of dimension n+1 consisting of homogeneous polynomials of degree n, that is, the subspace spanned by  x^n, \ x^{n-1}y, \ ..., \ y^n . Let P and Q be linear transformations on  V_{n} defined for f in  V_{n} by

     Pf=x \frac{ \partial f}{ \partial y} and  Qf=y \frac{\partial f}{\partial x}

    Find the minimal polynomial of (PQ-QP).

    Now, if my calculations are correct, for an arbitrary basis vector of the form  x^{n-k}y^k ,  (PQ-QP)(x^{n-k}y^k)=(n-2k)x^{n-k}y^k .

    Now, the minimal polynomial (denote m) is the unique monic polynomial of least degree s.t. m(PQ-QP)=0 (0 transformation). So in particular, I want  m((PQ-QP)(x^{n-k}y^k))=m((n-2k)x^{n-k}y^k)=0 , and moreover any linear combination of  (n-2k)x^{n-k}y^k to be equal to zero. How am I supposed to achieve this?

    Thanks a lot for all the help.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by Mimi89 View Post
    Hi everyone,

    could you please help me with the following?

    Let  n \geq 1 and let  V_{n} be the subspace of  \mathbb{R} [x,y] of dimension n+1 consisting of homogeneous polynomials of degree n, that is, the subspace spanned by  x^n, \ x^{n-1}y, \ ..., \ y^n . Let P and Q be linear transformations on  V_{n} defined for f in  V_{n} by

     Pf=x \frac{ \partial f}{ \partial y} and  Qf=y \frac{\partial f}{\partial x}

    Find the minimal polynomial of (PQ-QP).

    Now, if my calculations are correct, for an arbitrary basis vector of the form  x^{n-k}y^k ,  (PQ-QP)(x^{n-k}y^k)=(n-2k)x^{n-k}y^k .

    Now, the minimal polynomial (denote m) is the unique monic polynomial of least degree s.t. m(PQ-QP)=0 (0 transformation). So in particular, I want  m((PQ-QP)(x^{n-k}y^k))=m((n-2k)x^{n-k}y^k)=0 , and moreover any linear combination of  (n-2k)x^{n-k}y^k to be equal to zero. How am I supposed to achieve this?

    Thanks a lot for all the help.
    what you've got so far is correct. so \{n-2k, \ 0 \leq k \leq n \} is the set of eigenvalues of PQ-QP. since these eigenvalues are pairwise distinct, the characteristic and minimal polynomials of

    PQ-QP are equal. thus the minimal polynomial of PQ-QP is \prod_{k=0}^n (x-n+2k).
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