For example, $V(\vec x) = \frac{1}{2}(x_1^2+x_1x_2+x_2^2)$ has an equivalent representation $V(\vec x) = \frac {1}{2} x^TPx $ where $P$ is some matrix
Can someone make this connection clearer for me in the following three aspects:
What can we say about the definiteness of V and its matrix P (Does V and P always have the same definiteness?)
What are some properties of this matrix P (for any V quadratic not limited to the case at hand)?
Are there any physical intuition to quadratic forms? Does the polynomial and its associate matrix representation of some physical quantity?
The matrix can be taken to be symmetric, with $P_{ii}$ the coefficient of $x_i^2$, and $P_{ij} = P_{ji}$ half the coefficient of $x_i x_j$, in the quadratic form. Thus in your example, $P = \pmatrix{1 & 1/2 \cr 1/2 & 1\cr}$. The matrix has the same definiteness (positive definite, positive semidefinite, indefinite, negative semidefinite, negative definite) as the quadratic form.