For a real symmetric positive definite matrix \(A\) and \(v\in\mathbb R^n\), show that \[ \int_{\mathbb R^n}\exp(-x^TAx+2v^Tx)\,dx =\frac{\pi^{n/2}}{\sqrt{\det A}}\exp(v^TA^{-1}v). \] You may assume that \(\int_{-\infty}^{\infty}e^{-x^2}\,dx=\sqrt\pi\).
Solution
Complete the square, orthogonally diagonalize A, change variables, and integrate.