Positive definiteness

I was looking over the problem sheets for the course, and got stuck on sheet 8. Why is it when you are trying to find whether an inner product is a positive definite form for

v^t(1 2 3,2 3 4,3 4 10)w

you

a) have to find the canonical form and

b)that canonical form then implies whether or not the inner product is a positive definite form or not?

I would greatly appreciate your help

It is important to notice that the property of being positive definite for a bilinear form does not at all depend on a matrix representation. We say that the form $\langle \cdot, \cdot \rangle$ is positive definite if $\langle v, v\rangle >0$ for all $v \neq 0$. This definition does not refer to any matrix at all. Therefore, if you think about it carefully, you can check for positive definiteness using any matrix representative. You might say that one of the main reasons for coming up with the abstract notion of a bilinear form (or other abstract notions in linear algebra) is to naturally formulate properties or define quantities that are independent of bases, and for which computations or verifications can be carried out then in any convenient basis. I showed in class easy computations of determinants that exploit this principle.

On the other hand, it is instructive to note what’s involved purely in terms of matrices as well:

If matrices $A$ and $B$ are similar, then $A$ is positive definite if and only if $B$ is.

Since similarity is an equivalence relation, obviously, we need only check the implication

$A$ is positive definite $\Rightarrow$ $B$ is positive definite.

To see this, note that there is an invertible matrix $P$ such that

$B=P^tAP$

But then, for any non-zero $v$, we have

$v^tBv=v^tP^tAPv=(Pv)^tA(Pv)>0$

since $Pv\neq 0$ and $A$ is positive definite. So $B$ is positive definite.

Now, for a symmetric matrix in canonical form, I hope you can see why checking for positive definiteness is completely trivial.