The Multivariate guassian is given by:
$\mathcal{N}(\mathbf{x} ; \boldsymbol{\mu}, \mathbf{Q})=\frac{1}{\sqrt{|2 \pi \mathbf{Q}|}} \exp \left(-\frac{1}{2}(\mathbf{x}-\boldsymbol{\mu})^{T} \mathbf{Q}^{-1}(\mathbf{x}-\boldsymbol{\mu})\right)$
Where $Q^{-1}$ is the inverse of the covariance matrix. I know this is a quadratic form that measures the distance of x from the mean but why is it the inverse of the covariance matrix? Should'nt it be just the covariance matrix?
If there is some data matrix $A$, and the columns are mean subtracted then the distance from the origin of $Ax$ is: $<Ax,Ax>=x^TA^TAx= x^TDx$
where $D$ is the covariance matrix. So why does the inverse occur in the guassian? Can someone properly illustrate why inverse is needed? Is it related to the mahalanobis distance?