# Completing the square

• Entertainment

## Homework Statement

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

a) $$h(x,y) = xy$$
b) $$h(x,y,z) = yz$$
c) $$h(x,y,z) = x^2 + y^2 + z^2 + 2xy + 2xz + yz$$

## 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 $$h(x,y,z) = x^2 + 2xy + 2xz + y^2 + yz + z^2 = x^2 + 2x(y+z) + y^2 + yz + z^2 = (x+y+z)^2 - (y+z)^2 + y^2 + yz + z^2 = (x+y+z)^2 - yz$$, but I suspect that would require a solution of b) to get further.

Can you write $$xy$$ as a sum of quadratic terms? Hint: You might have to square the sum of two things.

Can you write $$xy$$ 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) $$xy = (x+\dfrac{y}{4})^2 - (x-\dfrac{y}{4})^2$$
Signature: (1,-1)

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

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

Am I overlooking something?

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.