I am trying to work out the polynomial expression in multivariate normal distribution. The inner product inside the epsilon for univariate (assuming 0 mean and unit variances/covariances for simplicity) is as follows:
$$p(x)\approx \exp\ \left(-\frac{1}{2}(x^2)\right)$$ so basically the concerned polynomial is $x^2$ For bivariate case the concerned polynomial is: $$\left(x^2\ -2xy\ + y^2\right)$$ This is the expanded form of $(x-y)^2$. Now for trivariate case, the concerned polynomial is: $$\left(x^2\ + y^2+z^2 -2xy\ -2xz -2yz\right)$$
This is now bit confusion as what is the factored form for this? Also in general what is the rule if there are more variables? There is a matrix form for it which requres inverse of the matrix but I am looking for polynomial case for intuitive understanding.