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Problem reducing quadratic to diagonal form

  • Thread starter lep11
  • Start date
380
7
1. Homework Statement
Reduce ##xy+zy## to diagonal form.

2. 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'.

3. 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.
 
Last edited:

lurflurf

Homework Helper
2,417
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hint
$$xy=\left(\frac{x}{2}+\frac{y}{2}\right)^2-\left(\frac{x}{2}-\frac{y}{2}\right)^2$$
 
380
7
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

Homework Helper
2,417
122
sorry try
$$\left( \frac{x}{2} \pm \frac{y}{\sqrt{2}}+\frac{z}{2} \right)^2$$
As the two
 

pasmith

Homework Helper
1,733
408
1. Homework Statement
Reduce ##xy+zy## to diagonal form.

2. 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'.

3. 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 [itex]u= x + z[/itex] so that [itex]Q = uy[/itex]? Doesn't the identity in post #2 then work?
 

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