# Is there an intuition about the matrix operations in the exponent of the multivariate normal distribution?

In the exponent of the multivariate distribution, there are 2 vectors and a square matrix multiplied together to get a scalar result:

$$(\mathbf{x} - \mu)^{\text{T}}\Gamma^{-1}(\mathbf{x} - \mu)$$

where $$\Gamma$$ is the covariance matrix of random variables $$X_1, ... X_n$$, $$\mathbf{x} = (x_1, ..., x_n)^{\text{T}}$$ is the vector of values $$X_1, ..., X_n$$ takes, and $$\mu = (\mu_1, ..., \mu_n)^\text{T}$$ is the vector of means for $$X_1, ..., X_n$$.

Right-multiplying a matrix by a vector, then left-multiplying the result by the transpose of that same vector seems like something which would have a nice intuitive explanation. Is there an intuitive explanation for these operations in general, and specifically for this case in the exponent of the multivariate normal distribution?

That quadratic form in the exponent is a squared Mahalanobis distance, so a measure of the distance between $$x$$ and the expected value $$\mu$$. For details see for instance Bottom to top explanation of the Mahalanobis distance?.