Matrix of a quadratic form and normal form.

Question

Let $\langle,\rangle$ be a symmetric bilinear form.Then we define the associated quadratic form $q:V\to \mathbb F$ by $q(x)=\langle x,x\rangle$ .We can also define a symmmetric bilinear form from a quadratic form by the polarisation identity $\langle x,y\rangle=\frac{1}{4}(q(x+y)-q(x-y))$.My question is what is the matrix of a quadratic form?Is it the matrix of the corresponding bilinear form obtained by polarisation?What is the simplest form called normal form of a quadratic form and how to reduce a quadratic form to its normal form.Does there exists an invertible matrix $P$ such that $P^tAP$ is diagonal?