multidimensional Gaussian integral

What if instead of one bell curve you have a smooth hill of probability over many variables at once — a Gaussian in n dimensions? The multidimensional Gaussian integral evaluates the integral over all of n-dimensional space of e^(-(1/2) x^T A x), where x is a vector of n variables and A is a symmetric positive-definite matrix that tilts and stretches the hill. The remarkable result packages all of that geometry into a single clean formula.

The answer is (2pi)^(n/2) divided by the square root of the determinant of A — written sqrt((2pi)^n / det A). You get there by diagonalizing A: a symmetric positive-definite matrix can be rotated to principal axes where the quadratic form splits into a sum of independent one-variable Gaussians, one per eigenvalue. The integral then factors into n ordinary Gaussian integrals, whose product carries a 1/sqrt(eigenvalue) each; multiplying the eigenvalues reconstructs det A. The rotation has determinant 1, so it does not change the volume — that is why only the determinant of A survives. Adding a linear term in the exponent shifts the peak and contributes a factor by completing the square.

This formula is the cornerstone of multivariate statistics (the normalization of the multivariate normal distribution), of equilibrium statistical mechanics, and especially of quantum and statistical field theory, where the free (Gaussian) path integral is exactly this with A a differential operator and det A a functional determinant. The condition that A be positive-definite is not a technicality: if any eigenvalue is non-positive the hill does not fall off in that direction and the integral diverges, so the determinant formula only holds when every eigenvalue is strictly positive.