Boundedness of a Quadratic Form

[thegreenlaser]

Given:
$$H_0 = x^2 + y^2 + 2axy$$

How does one go about finding the bounds on x and y, based on a? The author of a book I'm reading says that bounds are simple to show based on the different conditions |a|<1, |a|>1, or |a|=1. Unfortunately, I'm not finding it so simple, except for the |a|=1 case. Could someone at least point me in the right direction?

Thanks

 

[pmsrw3]

Given:
$$H_0 = x^2 + y^2 + 2axy$$

How does one go about finding the bounds on x and y, based on a? The author of a book I'm reading says that bounds are simple to show based on the different conditions |a|<1, |a|>1, or |a|=1. Unfortunately, I'm not finding it so simple, except for the |a|=1 case. Could someone at least point me in the right direction?

Thanks

Hmm. Does it help to write H₀ as

$$\alpha(x+y)^2 + \beta(x-y)^2$$

?

 

[thegreenlaser]

Hmm. Does it help to write H₀ as

$$\alpha(x+y)^2 + \beta(x-y)^2$$

?

I'm trying that now. It turns out to be:

$$\frac{1}{2}(1+a)(x+y)^2 + \frac{1}{2} (1-a)(x-y)^2$$

I have yet to see if that helps me in any way.

Also, just in case it was a little ambiguous in my first post, the goal is to find out whether if H₀ is bounded then x and y are bounded as well. If they are bounded, I need to know how they're bounded in terms of H₀

 

[pmsrw3]

In your OP, you said you were trying to find bounds on x and y. Does that mean that H₀ is a fixed number (presumably >0)?

 

[thegreenlaser]

^Just edited that in my last post. Yes, sorry, that was a little unclear.

 

[pmsrw3]

Ah! In that case, I think the answer is clear. If |a|<1, then you have a sum of two positive squares. Obviously those are both going to be bounded, and that will bound x and y. If |a|>1, you have a positive and a negative square, and you can make one as big as you like as long as you compensate by making the other big, too.

 

[thegreenlaser]

Ah! In that case, I think the answer is clear. If |a|<1, then you have a sum of two positive squares. Obviously those are both going to be bounded, and that will bound x and y. If |a|>1, you have a positive and a negative square, and you can make one as big as you like as long as you compensate by making the other big, too.

That does make sense. Thanks!

Now is there any way to do that quantitatively? i.e. Can we bound x and y directly in terms of H₀?

EDIT:
Breaking it into it's $$H_0 =\frac{1}{2}(1+a)(x+y)^2 + \frac{1}{2} (1-a)(x-y)^2$$ form did give me a lot of useful information about where the zeroes are and where it's positive/negative, but I still haven't coaxed any boundedness info out of it.