Positive definite matrix bounded below

[Charles49]

Homework Statement

Let A be a positive definite n\times n real matrix, b\in\mathbb{R}^n, and consider the quadratic polynomial Q(x)=\frac{1}{2}\langle{x, Ax\rangle}-\langle{b, x\rangle}. Show that Q is bounded below.

2. The attempt at a solution

I have to come up with a constant m so that Q(x)\ge m for all x\in\mathbb{R}^n. I see that Q looks a lot like a parabola. I know how to find the lower bound of a parabola opening upward but I don't know how to generalize this to quadratic forms.

 

[micromass]

What is a??

 

[Charles49]

Sorry, its supposed to be x.

 

[micromass]

Let C be such that C^2=A (why does C exist?) and let c=C^{-1}b (why is C invertible?).

Calculate

\frac{1}{2}<Cx-c,Cx-c>

 

[Charles49]

This is what I got but I don't know what it does

\frac{1}{2}\langle{Cx-C^{-1}b, Cx-C^{-1}b\rangle} = \frac{1}{2}||Cx-C^{-1}b||\\<br /> =\frac{1}{2}||C^{-1}(Ax-b)||

C exists because A is positive definite. C^{-1}[\tex] exists because A has no zero eigenvalues.

 

[micromass]

Yes, so that shows that

&lt;Cx-C^{-1}b,Cx-X^{-1}b&gt;\geq 0

Now try to evaluate this in another way... Use the linearity of the inproduct.

 

[Charles49]

This suggestion helped a lot. Thanks.