Find Inner Product for Quadratic Form in R^3

[Denis99]

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?

 

[StoneTemplePython]

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

 

[Denis99]

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
https://en.m.wikipedia.org/wiki/Inner_product_space
(in part Definition)

 

[StoneTemplePython]

I have to work with definition like this one from definition of inner space in here
https://en.m.wikipedia.org/wiki/Inner_product_space
(in part Definition)

But I can't figure out how you could understand that and say this

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 $A^\frac{1}{2}$