Positive definit quadratic form

Question

I have a very elementar question...

If I have a quadratic form $$ T(x_1,x_2)=x^TQx $$ and $Q$ is a positive definit symmetric 2x2-matrix, then does this mean that the quadratic form is positive definit, i.e. $$ T(x_1,x_2)=x^TQx>0? $$

Answer 1

If you mean $x=[x_1, x_2]$ then, yes, if for $x \not = 0$ $x^T Q x >0$, the $Q$ is positive definite. That is by definition!

Answer 2

Yes, and here are the rather easy highlights to prove this:

$$Q=\begin{pmatrix}a&b\\b&c\end{pmatrix}\;\;\text{is definite positive}\;\iff\begin{cases} a>0\\{}\\ac-b^2>0\end{cases}$$

Now check that for any $\;\underline x\in\Bbb R^2\;$ this means $\;\underline x^tQ\underline x>0\;$

BTW, such a quadratic form defines an inner product iff $\;Q\;$ is positive definite, precisely...