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Thread: Finding minimal polynomial

  1. #1
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    Finding minimal polynomial

    Hi everyone,

    could you please help me with the following?

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

    $\displaystyle Pf=x \frac{ \partial f}{ \partial y} $ and $\displaystyle 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 $\displaystyle x^{n-k}y^k $, $\displaystyle (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 $\displaystyle m((PQ-QP)(x^{n-k}y^k))=m((n-2k)x^{n-k}y^k)=0 $, and moreover any linear combination of $\displaystyle (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
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    Quote Originally Posted by Mimi89 View Post
    Hi everyone,

    could you please help me with the following?

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

    $\displaystyle Pf=x \frac{ \partial f}{ \partial y} $ and $\displaystyle 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 $\displaystyle x^{n-k}y^k $, $\displaystyle (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 $\displaystyle m((PQ-QP)(x^{n-k}y^k))=m((n-2k)x^{n-k}y^k)=0 $, and moreover any linear combination of $\displaystyle (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 $\displaystyle \{n-2k, \ 0 \leq k \leq n \}$ is the set of eigenvalues of $\displaystyle PQ-QP.$ since these eigenvalues are pairwise distinct, the characteristic and minimal polynomials of

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