Completing the square

  • #1

Homework Statement



Complete the square to determine the signature of the following quadratic forms:

a) [tex]h(x,y) = xy[/tex]
b) [tex]h(x,y,z) = yz[/tex]
c) [tex]h(x,y,z) = x^2 + y^2 + z^2 + 2xy + 2xz + yz[/tex]

Homework Equations





The Attempt at a Solution



I suspect there's a really simple trick to this, but I can't seem to figure it out at the moment.

For c), we get [tex]h(x,y,z) = x^2 + 2xy + 2xz + y^2 + yz + z^2 = x^2 + 2x(y+z) + y^2 + yz + z^2 = (x+y+z)^2 - (y+z)^2 + y^2 + yz + z^2 = (x+y+z)^2 - yz[/tex], but I suspect that would require a solution of b) to get further.
 

Answers and Replies

  • #2
fzero
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Can you write [tex]xy[/tex] as a sum of quadratic terms? Hint: You might have to square the sum of two things.
 
  • #3
Can you write [tex]xy[/tex] as a sum of quadratic terms? Hint: You might have to square the sum of two things.
After messing around a bit, I found that

a) [tex]xy = (x+\dfrac{y}{4})^2 - (x-\dfrac{y}{4})^2[/tex]
Signature: (1,-1)

b) Similar to a): [tex]yz = (y+\dfrac{z}{4})^2 - (y-\dfrac{z}{4})^2[/tex]
Signature: (1,-1,0)

c) Continuing from before, and using b); [tex](x+y+z)^2 - yz = (x+y+z)^2 + (y-\frac{z}{4})^2 - (y+\frac{z}{4})^2[/tex]
Signature: (1,1,-1)

Am I overlooking something?
 
  • #4
fzero
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The signature of a quadratic form is usually defined as the number of positive eigenvalues minus the number of negative eigenvalues of the corresponding matrix. You can compute this from your results.
 

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