= Gaussian quadratic exponential moment
{c}
{title2=$\mathbb E e^{\langle b,Z\rangle+\langle Z,QZ\rangle}$}
For a real standard <Gaussian random vector> $Z$, a real symmetric matrix $Q$, and $I-2Q$ strictly positive,
$$
\mathbb E e^{b^TZ+Z^TQZ}=\det(I-2Q)^{-1/2}\exp\left[\frac12b^T(I-2Q)^{-1}b\right].
$$
Diagonalize $Q$ and complete the square in each coordinate. Failure of strict positivity makes the positive exponential integral divergent. The same formula extends to a self-adjoint trace-class quadratic form of an isonormal <Gaussian process>, with square-summable linear coefficients and a <Fredholm determinant>. A linear-plus-quadratic <Gaussian random field> functional is reduced to this form by its covariance square root.
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