I am reading a paper and I am a bit confused by a paragraph in it.
The paper introduces early on the formula for multidimensional Gaussians:
$$c\cdot e^{-\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)}$$
Where $x$ is an $n$ dimensional vector, $\Sigma$ is an $n \times n$ covariance martix, and $\mu$ is the $n$ dimensional mean of the distribution.
Some time later the paper claims that this formula can be expressed as a dot product between a quadratic vector basis $b(x)$ and a coefficient vector $q_i$:
$$ c\cdot e^{-\frac{1}{2}b(x)^Tq_i} $$
I understand the concept of expressing polynomials in terms of linear algebra. For example it's easy to see that any quadratic polynomial is a dot product of a vector basis and a coefficient vector e.g:
$[x^2,x,1] \cdot [a,b,c] = ax^2+bx+c$
However I am unable to take the exponent of the first formula and rewrite it as a product of a quadratic vector and a coefficient vector as the paper claims. The main thing is, I need to find the explicit representation of the $q_i$ vector but I am not being able to figure out the algebra for the transformation.