= Linear-source multivariate Gaussian integral
{title2=$\int_{\mathbb R^n}e^{-u^TAu/2+b^Tu}\,du=(2\pi)^{n/2}(\det A)^{-1/2}e^{b^TA^{-1}b/2}$}
For a real <symmetric positive-definite matrix> $A$ and a real <vector> $b\in\mathbb R^n$, <completing the square> with $v=u-A^{-1}b$ gives a real <change of variables>. Orthogonal diagonalization reduces the remaining integral to the product of one-dimensional <Gaussian integrals>, proving the displayed expression. Positivity ensures convergence and the positive determinant square root. Replacing matrices by regulated field kernels gives the <Gaussian functional integral>; an oscillatory Minkowski integral instead requires a causal convergence prescription.
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