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  1. MHB Apprentice

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    #1
    Let $ \displaystyle <x, x>=3x_{1}^2+2x_{2}^2+x_{3}^2-4x_{1}x_{2}-2x_{1}x_{3}+2x_{2}x_{3} $ be a quadratic form in V=R, where $ \displaystyle x=x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3}$ (in the base $ \displaystyle {e_{1},e_{2},e_{3}}$.
    Find the inner product corresponding to this quadratic form.

    Is this that easy that you have to change '' second'' x-es for y (for example to write $ \displaystyle 2x_{2}y_{3}$ instead of $ \displaystyle 2x_{2}x_{3}$ at the end), or what I have to do?
    Last edited by Denis99; June 3rd, 2019 at 15:53.

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  3. MHB Apprentice

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    #2
    Quote Originally Posted by Denis99 View Post
    Let $ \displaystyle <x, x>=3x_{1}^2+2x_{2}^2+x_{3}^2-4x_{1}x_{2}-2x_{1}x_{3}+2x_{2}x_{3} $ be a quadratic form in V=R, where $ \displaystyle x=x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3}$ (in the base $ \displaystyle {e_{1},e_{2},e_{3}}$.
    Find the inner product corresponding to this quadratic form.

    Is this that easy that you have to change '' second'' x-es for y (for example to write $ \displaystyle 2x_{2}y_{3}$ instead of $ \displaystyle 2x_{2}x_{3}$ at the end), or what I have to do?
    what definition do you have to work with here? If it were me I'd iterate to the result-- start by encoding this with standard basis vectors, i.e.

    $\langle x, x \rangle =3x_{1}^2+2x_{2}^2+x_{3}^2-4x_{1}x_{2}-2x_{1}x_{3}+2x_{2}x_{3} = \mathbf x^T A \mathbf x$

    where $A$ is real symmetric positive definite. Then take the square root of $A$ and let that be your basis i.e. consider
    $ A^\frac{1}{2} \mathbf x$
    where the $kth$ column of $ A^\frac{1}{2} $ is denoted by $e_k$ in your text

  4. MHB Apprentice

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    #3 Thread Author
    Quote Originally Posted by steep View Post
    what definition do you have to work with here? If it were me I'd iterate to the result-- start by encoding this with standard basis vectors, i.e.

    $\langle x, x \rangle =3x_{1}^2+2x_{2}^2+x_{3}^2-4x_{1}x_{2}-2x_{1}x_{3}+2x_{2}x_{3} = \mathbf x^T A \mathbf x$

    where $A$ is real symmetric positive definite. Then take the square root of $A$ and let that be your basis i.e. consider
    $ A^\frac{1}{2} \mathbf x$
    where the $kth$ column of $ A^\frac{1}{2} $ is denoted by $e_k$ in your text
    I have to work with definition like this one from definition of inner space in here

    (in part Definition)

  5. MHB Apprentice

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    #4
    Quote Originally Posted by Denis99 View Post
    I have to work with definition like this one from definition of inner space in here

    (in part Definition)
    But I can't figure out how you could understand that and say this

    Quote Originally Posted by Denis99 View Post
    Find the inner product corresponding to this quadratic form.

    Is this that easy that you have to change '' second'' x-es for y (for example to write $ \displaystyle 2x_{2}y_{3}$ instead of $ \displaystyle 2x_{2}x_{3}$ at the end), or what I have to do?
    The reality is that one way or another you need to find $A^\frac{1}{2}$

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