Problem reducing quadratic to diagonal form

[lep11]

Homework Statement

Reduce $xy+zy$ to diagonal form.

Homework Equations

The desired diagonal form is $Q(\vec{x})=(\alpha_1(\vec{x}))^2+...+(\alpha_k(\vec{x}))^2-(\alpha_{k+1}(\vec{x}))^2-...-(\alpha_{k+l}(\vec{x}))^2,$ where $\alpha_i$ are linearly independent linear functions. Also known as 'changing the variables in quadratic form'.

The Attempt at a Solution

$xy+zy=y^2+yx+yz+(\frac{x}{2})^2+\frac{xz}{2}+(\frac{z}{2})^2-((\frac{x}{2})^2+\frac{xz}{2}+(\frac{z}{2})^2)-y^2=(y+\frac{x}{2}+\frac{z}{2})^2+(\frac{x}{2}+\frac{z}{2})^2-y^2$, but now
$\alpha_1=y+\frac{x}{2}+\frac{z}{2}$
$\alpha_2=\frac{x}{2}+\frac{z}{2}$
$\alpha_3=y$
are linearly dependent. I have tried several different substitutions without success. There's also the matrix method which I am not familiar with. This rather simple problem is giving me headache.

 

[lurflurf]

hint
$$xy=\left(\frac{x}{2}+\frac{y}{2}\right)^2-\left(\frac{x}{2}-\frac{y}{2}\right)^2$$

 

[lep11]

hint
$$xy=\left(\frac{x}{2}+\frac{y}{2}\right)^2-\left(\frac{x}{2}-\frac{y}{2}\right)^2$$

Using that lead to situation where the $\alpha$'s weren't linearly independent.

 

[lurflurf]

sorry try
$$\left( \frac{x}{2} \pm \frac{y}{\sqrt{2}}+\frac{z}{2} \right)^2$$
As the two

 

[pasmith]

Homework Statement

Reduce $xy+zy$ to diagonal form.

Homework Equations

The desired diagonal form is $Q(\vec{x})=(\alpha_1(\vec{x}))^2+...+(\alpha_k(\vec{x}))^2-(\alpha_{k+1}(\vec{x}))^2-...-(\alpha_{k+l}(\vec{x}))^2,$ where $\alpha_i$ are linearly independent linear functions. Also known as 'changing the variables in quadratic form'.

The Attempt at a Solution

$xy+zy=y^2+yx+yz+(\frac{x}{2})^2+\frac{xz}{2}+(\frac{z}{2})^2-((\frac{x}{2})^2+\frac{xz}{2}+(\frac{z}{2})^2)-y^2=(y+\frac{x}{2}+\frac{z}{2})^2+(\frac{x}{2}+\frac{z}{2})^2-y^2$, but now
$\alpha_1=y+\frac{x}{2}+\frac{z}{2}$
$\alpha_2=\frac{x}{2}+\frac{z}{2}$
$\alpha_3=y$
are linearly dependent. I have tried several different substitutions without success. There's also the matrix method which I am not familiar with. This rather simple problem is giving me headache.

What happens if you set $u= x + z$ so that $Q = uy$? Doesn't the identity in post #2 then work?